ResearchMath-14K: Scaling Research-Level Mathematics via Agents
Paper • 2605.28003 • Published • 47
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2408.03803#0 | 2408.03803 | https://arxiv.org/abs/2408.03803 | math.NT | conjecture | critical_threshold | D | universality | bound_improvement | null | Let ε* be the infimum of the ε > 0 for which there exists δ > 0 such that, for every B ≥ 1 and every nonzero integer a, the sum over integers m ≤ x^(1−ε) with P⁺(m) ≤ x^δ and (m,a) = 1 of |π(x;m,−a) − π(x)/φ(m)| is ≪_(B,ε,a) x/(log x)^B. Here P⁺(m) is the largest prime factor of m, with P⁺(1) = 0; π(x) counts primes p ... | Give the infimum as one number; ε* = 0 is equivalent to Hypothesis Z(1) holding. | 0 | true | Does Hypothesis Z(1) hold: for every epsilon > 0, does there exist delta > 0 such that, for every B >= 1 and every nonzero integer a, one has sum over integers m <= x^(1-epsilon) with P^+(m) <= x^delta and (m,a)=1 of |pi(x;m,-a) - pi(x)/phi(m)| <<_(B,epsilon,a) x/(log x)^B, where P^+(m) is the largest prime factor of m... | openai/gpt-6-sol |
2511.15050#3 | 2511.15050 | https://arxiv.org/abs/2511.15050 | math.AP | conjecture | critical_threshold | D | universality | bound_improvement | generalization | Consider the three-dimensional incompressible Navier–Stokes system ∂_t u+(u·∇)u−(∂_{x1}²+∂_{x2}²)u=−∇p, div u=0, on [0,1]_{x1}×ℝ²_{(x2,x3)} for t>0, with u=0 at x1=0,1. For ρ>0, σ≥1 and N∈ℤ_{≥0}, define G_{ρ,σ,N} by the squared norm Σ_{m≥0} L_{ρ,m}²‖∂_{x3}^m A‖_{H^N}², where L_{ρ,m}=ρ^{m+1}(m+1)^{6+2σ}/(m!)^σ. Suppose ... | The answer is the supremum defined in the question; 0 means instantaneous x2-analyticity holds for every admissible datum at every σ≥1. | 0 | true | Consider the three-dimensional incompressible anisotropic Navier-Stokes system with dissipation only in the horizontal variables, dt u + (u . grad) u - (d_{x1}^2 + d_{x2}^2) u = - grad p, div u = 0, posed for t > 0 on the strip [0,1]_{x1} x R^2_{(x2,x3)} with the no-slip boundary condition u = 0 on x1 = 0 and x1 = 1 an... | openai/gpt-6-sol |
2407.17820#5 | 2407.17820 | https://arxiv.org/abs/2407.17820 | math.NT | conjecture | critical_threshold | D | universality | bound_improvement | null | Let rad(m) be the product of the distinct prime divisors of the positive integer m. What is the supremum of the set consisting of 0 and all numbers ε/(1+ε) for which ε>0 and infinitely many pairwise coprime positive-integer triples (a,b,c) satisfy a+b=c>rad(abc)(log rad(abc))^ε? | Give one real number; 0 means that for every ε>0, only finitely many such triples exist. | 0 | true | Let rad(m) be the product of the distinct prime divisors of the positive integer m. The abc conjecture asserts that, for every t>1, only finitely many pairwise coprime positive-integer triples (a,b,c) satisfy a+b=c>rad(abc)^t. Does the following logarithmic weakening hold: for every \varepsilon>0, are there only finite... | openai/gpt-6-sol |
2604.25177#0 | 2604.25177 | https://arxiv.org/abs/2604.25177 | math.NT | conjecture | critical_threshold | D | existence | bound_improvement | null | Fix a nonzero integer a. Let δ* be the supremum of {0} together with all δ>0 for which there exists A>0 such that, for positive M,N,X with MN=X and every modulus scale Q≤X^{1/2+δ}, one has
∑_{q∼Q,(a,q)=1} |∑_{n∼N,m∼M,mn≡a (mod q)} μ(m)log n − (1/φ(q))∑_{n∼N,m∼M,(mn,q)=1} μ(m)log n| ≪ X/(log X)^A.
What is δ*? | Report the supremum as one number for the fixed a; δ*>0 means the original existence claim holds. | null | false | Fix a nonzero integer a. For positive parameters M,N,X with MN=X, let α_m=μ(m) for m∼M and β_n=log n for n∼N. Does there exist a fixed δ>0 and an exponent A>0 such that, for every modulus scale Q≤X^{1/2+δ}, one has
sum_{q∼Q,(a,q)=1} | sum_{n∼N,m∼M,mn≡a (mod q)} μ(m)log n − (1/φ(q)) sum_{n∼N,m∼M,(mn,q)=1} μ(m)log n | ≪ ... | openai/gpt-6-sol |
2411.12567#1 | 2411.12567 | https://arxiv.org/abs/2411.12567 | math.NT | conjecture | critical_threshold | D | universality | bound_improvement | asymptotic_rate | Let \Gamma be a cocompact Fuchsian group acting on the hyperbolic upper half-plane \mathbb{H}, and let l be a fixed closed geodesic segment whose stabilizer is \Gamma_1=\langle\gamma_1\rangle, with \gamma_1 a primitive hyperbolic element of \Gamma. For R>0 put X=\cosh R>1, and let N(X,l) count the double cosets \gamma\... | Give the infimum as one number; the original bound holds for every \epsilon>0 exactly when this number is 0. | 0 | true | Let \Gamma be a cocompact Fuchsian group acting on the hyperbolic upper half-plane \mathbb{H}, and let l be a fixed closed geodesic segment whose stabilizer is \Gamma_1=\langle\gamma_1\rangle, with \gamma_1 a primitive hyperbolic element of \Gamma. For R>0 put X=\cosh R>1 and define N(X,l) to be the number of double co... | openai/gpt-6-sol |
2510.09028#0 | 2510.09028 | https://arxiv.org/abs/2510.09028 | math.ST | conjecture | critical_threshold | D | comparison | bound_improvement | asymptotic_rate | Fix T>0, alpha in (1/2,1), and integers d,r>=1. Let B be an r-dimensional Brownian motion, and let X be an R^d-valued solution on [0,T] of X_t=X_0+int_0^t K(t-s)a(X_s)dB_s+int_0^t K(t-s)b(X_s)ds, where a:R^d->R^{d x r} and b:R^d->R^d are globally Lipschitz. Assume K is C^1 on (0,infinity) and, for every u in (0,T], |K(... | The answer is the largest universally valid convergence exponent for this grid-based reconstruction bound. | 0.5 | true | Fix T>0, alpha in (1/2,1), d,r>=1, and an r-dimensional Brownian motion B. Let X be the R^d-valued solution on [0,T] of X_t=X_0+int_0^t K(t-s)a(X_s)dB_s+int_0^t K(t-s)b(X_s)ds, where a:R^d->R^{d x r} and b:R^d->R^d are globally Lipschitz. Assume K is C^1 on (0,infinity) and, for every u in (0,T], |K(u)|<=c u^{alpha-1}/... | openai/gpt-6-sol |
2605.15434#0 | 2605.15434 | https://arxiv.org/abs/2605.15434 | math.NT | conjecture | critical_threshold | D | universality | bound_improvement | asymptotic_rate | Let Q(x_1,x_2,x_3,x_4)=x_1x_4-x_2x_3 and, for every N∈N, let A(N)=|{(x_1,x_2,x_3,x_4)∈Z^4: Q(x_1,x_2,x_3,x_4)=1 and x_1^2+x_2^2+x_3^2+x_4^2≤N^2}|. Selberg's estimate is A(N)=6N^2+O(N^{4/3}). What is the critical value inf{ε>0: A(N)=6N^2+O_ε(N^{1+ε}) as N→∞}, where the implied constant may depend on ε? | The answer is the infimum of the positive exponents for which the stated bound holds. | 0 | true | Let Q(x_1,x_2,x_3,x_4)=x_1x_4-x_2x_3, and, for every N∈N, let A(N)=|{(x_1,x_2,x_3,x_4)∈Z^4: Q(x_1,x_2,x_3,x_4)=1 and x_1^2+x_2^2+x_3^2+x_4^2≤N^2}|. Selberg's estimate is A(N)=6N^2+O(N^{4/3}). Is it true that, for every ε>0, A(N)=6N^2+O_ε(N^{1+ε}) as N→∞, with the implied constant allowed to depend on ε? | openai/gpt-6-sol |
2411.01577#2 | 2411.01577 | https://arxiv.org/abs/2411.01577 | math.AP | conjecture | critical_threshold | D | universality | bound_improvement | null | Let λ>1, and let φ(x,ξ) be a smooth phase on R²×R satisfying rank((∂ξ∂xφ(x,ξ));(∂ξ²∂xφ(x,ξ)))=2. Set φ^λ(x,ξ)=λφ(x/λ,ξ) and T^λf(x)=∫_R exp(iφ^λ(x,ξ))a^λ(x,ξ)f(ξ)dξ, where a^λ is smooth and supported in x in B_λ:=B(0,λ). For each frequency interval θ with center ξ_θ and each v∈[0,λ], let γ_θ be determined locally by ∂ξ... | Give the critical exponent inf E; a value of 0 means the estimate holds for every ε>0. | 0 | true | Let lambda>1, let phi(x,xi) be a smooth phase on R^2 x R satisfying the Carleson--Sjoelin condition rank((partial_xi partial_x phi(x,xi));(partial_xi^2 partial_x phi(x,xi)))=2, set phi^lambda(x,xi)=lambda phi(x/lambda,xi), and define T^lambda f(x)=integral_R exp(i phi^lambda(x,xi)) a^lambda(x,xi)f(xi) dxi, where a^lamb... | openai/gpt-6-sol |
2601.03837#1 | 2601.03837 | https://arxiv.org/abs/2601.03837 | math.MG | conjecture | critical_threshold | D | existence | bound_improvement | existence_construction | What is the critical exponent \(q_* = \inf\{q>1:\text{there exist }n\in\mathbb N,\ k\in\{1,\ldots,n\},\text{ and a closed AD-}k\text{-regular }E\subset\mathbb H^n\text{ that satisfies the }q\text{-geometric lemma for the projection }\iota\text{-numbers but does not have BPLI}\}\), with \(\inf\varnothing=+\infty\)? The ... | Report the single critical exponent; \(q_*=1\) means that such a counterexample exists for every \(q>1\). | 1 | true | For every \(q>1\), does there exist \(n\in\mathbb N\), \(k\in\{1,\ldots,n\}\), and a closed AD-\(k\)-regular set \(E\subset\mathbb H^n\) (for the Koranyi distance \(d((z,t),(z',t'))=\|(z',t')^{-1}\cdot(z,t)\|\), \(\|(z,t)\|=(|z|^4+16t^2)^{1/4}\)) such that \(E\) satisfies the \(q\)-geometric lemma for the projection \(... | openai/gpt-6-sol |
2604.22711#3 | 2604.22711 | https://arxiv.org/abs/2604.22711 | math.NT | conjecture | critical_threshold | D | existence | bound_improvement | null | Fix a connected reductive algebraic group G over Q. For each positive integer N, let S_N={∞}∪{p: p is a prime dividing N}. For any Levi subgroup M of G, let U_M(Q) be the set of unipotent M(Q)-conjugacy classes, and for O_γ∈U_M(Q), let a^M(S_N,O_γ) be the global coefficient attached to this orbit in Arthur's fine geome... | Give the infimum as one value, using +∞ if no such b exists. | null | false | Fix a connected reductive algebraic group G over \mathbb{Q}. For every positive integer N, let S_N=\{\infty\}\cup\{p:\ p\text{ is a prime dividing }N\}. If M is any Levi subgroup of G, let \mathcal{U}_M(\mathbb{Q}) be the set of unipotent M(\mathbb{Q})-conjugacy classes, and, for \mathcal{O}_\gamma\in\mathcal{U}_M(\mat... | openai/gpt-6-sol |
2504.04107#2 | 2504.04107 | https://arxiv.org/abs/2504.04107 | math.AP | conjecture | critical_threshold | D | universality | bound_improvement | hypothesis_weakening | Let α>0, m₀∈H¹(T²;S²), W be a real Brownian motion, and g=fh:T²→R³, where f:T²→R and h∈R³ is constant. For the Stratonovich SLLG equation dm=[−m×Δm+α(Δm+|∇m|²m)]dt+(m×g)∘dW on T², what is the critical noise-regularity exponent σ for constructing weak solutions by Dirichlet-energy methods in the class |m|=1, m∈C([0,T];L... | Give the critical value of σ as one number. | 1 | true | Let alpha>0, m_0 in H^1(T^2;S^2), W be a real Brownian motion, and let g=fh:T^2->R^3 with f:T^2->R and h in R^3 constant, where g belongs to H^sigma(T^2;R^3). For the Stratonovich SLLG equation d m=[-m x Delta m+alpha(Delta m+|nabla m|^2m)]dt+(m x g) circle dW on T^2, is sigma=1 the sharp noise-regularity threshold for... | openai/gpt-6-sol |
1712.00405#3 | 1712.00405 | https://arxiv.org/abs/1712.00405 | math.CV | conjecture | critical_threshold | D | universality | structural_characterization | null | Let X range over compact connected Riemann surfaces, z_0 over points of X, and ω over Kähler forms with ∫_X ω=1. For each choice, let \tilde{\Phi} be the envelope of all π_X^*ω-plurisubharmonic functions Ψ on X×\overline{\mathbb D} such that \limsup_{\zeta\to\zeta'}Ψ(\zeta)≤0 for every \zeta'∈X×∂\mathbb D and \nu_{(z_0... | Give the supremum as one number; it equals 1 exactly when the original regularity assertion holds. | 1 | true | Let X be a compact connected Riemann surface, let z_0\in X, and let \omega be a Kähler form with \int_X\omega=1. Let \mathbb D be the unit disc, let \pi_X:X\times\overline{\mathbb D}\to X be projection, and define \tilde{\Phi} as the envelope of all \pi_X^*\omega-plurisubharmonic functions \Psi on X\times\overline{\mat... | openai/gpt-6-sol |
2411.16935#0 | 2411.16935 | https://arxiv.org/abs/2411.16935 | math.CA | conjecture | critical_threshold | D | existence | existence_construction | bound_improvement | Let D⊂R² be the unit disk. For a compact convex set X⊂R² and l≥0, let P_X(l) be the probability that a directed line segment of length l, whose initial point is chosen uniformly from X and whose direction is chosen uniformly from [0,2π), lies entirely in X. What is the supremum of {0}∪{δ>0 : for every compact convex no... | Give one real number; the original existence claim holds exactly when that number is positive. | null | false | Let D⊂R^2 be the unit disk. For a compact convex set X⊂R^2 and l≥0, let P_X(l) be the probability that a directed line segment of length l, whose initial point is chosen uniformly from X and whose direction is chosen uniformly from [0,2π), lies entirely in X. Does there exist a constant δ>0 such that, for every compact... | openai/gpt-6-sol |
2502.14602#2 | 2502.14602 | https://arxiv.org/abs/2502.14602 | math.AP | conjecture | critical_threshold | D | extension | generalization | hypothesis_weakening | Let Omega subset R^3 be a bounded C^{2,beta} domain, with beta in (0,1). For each alpha in (1,2), let Omega_epsilon be Omega minus periodically separated holes T_{epsilon,k}=x_{epsilon,k}+epsilon^alpha T_0, where T_0 is a bounded simply connected C^{2,beta} reference obstacle and the hole separation is of order epsilon... | Give the single real number alpha_*; alpha_*=2 means the stated estimate extends to every alpha in (1,2). | 2 | true | Let Omega subset R^3 be a bounded C^{2,beta} domain, with beta in (0,1), and let Omega_epsilon=Omega minus the union of periodically separated holes T_{epsilon,k}=x_{epsilon,k}+epsilon^alpha T_0, where T_0 is a bounded simply connected C^{2,beta} reference obstacle, the hole separation is of order epsilon, and alpha is... | openai/gpt-6-sol |
0707.1756#5 | 0707.1756 | https://arxiv.org/abs/0707.1756 | math.NT | conjecture | critical_threshold | D | universality | bound_improvement | null | Let E(T)=∫₀ᵀ|ζ(1/2+it)|²dt−T(log(T/(2π))+2γ−1), where γ is Euler’s constant, and let Δ(x)=∑_{n≤x}d(n)−x(log x+2γ−1), where d(n) counts the positive divisors of n. What is the infimum of real r≥0 such that, for every fixed real k>2 and every ε>0, both ∫ₜ²ᵀ|E(t+G)−E(t−G)|ᵏdt ≪_{k,ε} T^{1+ε}G^{k/2} and ∫ₜ²ᵀ|Δ(t+G)−Δ(t−G)|... | Give the critical lower-range exponent as one number. | 0 | true | Let E(T) = integral_0^T |zeta(1/2+it)|^2 dt - T(log(T/(2pi)) + 2gamma - 1), where gamma is Euler's constant, and let Delta(x) = sum_{n<=x} d(n) - x(log x + 2gamma - 1), where d(n) is the number of positive divisors of n. For every fixed real k>2 and every epsilon>0, prove that, uniformly as T tends to infinity for T^ep... | openai/gpt-6-sol |
1706.06712#15 | 1706.06712 | https://arxiv.org/abs/1706.06712 | math.NT | conjecture | critical_threshold | D | existence | bound_improvement | asymptotic_rate | Let M(x_1,x_2,x_3)=x_1^2+x_2^2+x_3^2-x_1x_2x_3, let V_{k,M}(Z)={x in Z^3:M(x)=k}, and let h(k) be the number of orbits of V_{k,M}(Z) under the group Gamma generated by coordinate permutations, changes of sign in two coordinates, and the Vieta involutions (x_1,x_2,x_3)->(x_2x_3-x_1,x_2,x_3) and its coordinate analogues.... | Give the supremum as one extended-real value; the original question has answer Yes exactly when that value is positive. | null | false | Let M(x_1,x_2,x_3)=x_1^2+x_2^2+x_3^2-x_1x_2x_3, let V_{k,M}(Z)={x in Z^3:M(x)=k}, and let h(k) be the number of orbits of V_{k,M}(Z) under the group Gamma generated by coordinate permutations, changes of sign in two coordinates, and the Vieta involutions (x_1,x_2,x_3)->(x_2x_3-x_1,x_2,x_3) and its coordinate analogues.... | openai/gpt-6-sol |
2410.23993#0 | 2410.23993 | https://arxiv.org/abs/2410.23993 | math.CA | conjecture | critical_threshold | D | existence | bound_improvement | existence_construction | What is the critical exponent p_* = inf{p in (1,infinity] : there exists C_p<infinity, depending only on p, such that for every dimension d>=1, every convex, bounded, closed, symmetric set G subset R^d with non-empty interior, and every f in L^p(R^d), ||sup_{t>0}|M_t^G f|||_{L^p(R^d)} <= C_p||f||_{L^p(R^d)}}? Here M_t^... | Give the single numerical value of p_*; p_*=1 corresponds to the original inequality holding for every p in (1,infinity]. | 1 | true | For each fixed p in (1,infinity], does there exist a constant C_p<infinity, depending only on p, such that for every dimension d>=1, every convex, bounded, closed, symmetric set G subset R^d with non-empty interior, and every f in L^p(R^d), the maximal inequality ||sup_{t>0}|M_t^G f|||_{L^p(R^d)} <= C_p||f||_{L^p(R^d)}... | openai/gpt-6-sol |
2408.10881#1 | 2408.10881 | https://arxiv.org/abs/2408.10881 | math.NT | conjecture | critical_threshold | D | existence | asymptotic_rate | bound_improvement | Let f(N) be the maximum cardinality of a set A⊆[N]={1,2,...,N} such that every solution (x,y,w,z)∈A⁴ of 2x+2y=3w+z is trivial, meaning x=y=w=z. What is the critical exponent ρ*=inf{ρ∈ℝ : f(N)=O(N^ρ) as N→∞}? | Report the numerical value of ρ*; the original existence assertion holds exactly when ρ*<1. | null | false | Let f(N) be the maximum cardinality of a set A subseteq [N]={1,2,...,N} such that every solution (x,y,w,z) in A^4 of 2x+2y=3w+z is trivial, i.e. x=y=w=z. Does there exist a fixed rho<1 such that f(N)=O(N^rho) as N tends to infinity? | openai/gpt-6-sol |
1706.06189#0 | 1706.06189 | https://arxiv.org/abs/1706.06189 | math.PR | conjecture | critical_threshold | D | universality | hypothesis_weakening | generalization | Fix M >= 0 and nonnegative integers nu_1,...,nu_M. For each n, let H be an n x n GUE matrix with external source B and density proportional to exp{-tr(H-B)^2}. Let G_j, j=1,...,M, be independent standard complex Ginibre matrices of size (nu_{j-1}+n) x (nu_j+n), with nu_0=0, all independent of H. Set W_M=G_M^*...G_1^*HG... | Give the single supremum t; a value of 1 means the assertion holds for every 0 <= a < 1. | 1 | true | Fix M >= 0 and nonnegative integers nu_1,...,nu_M. For each n, let H be an n x n GUE matrix with external source B, having density proportional to exp{-tr(H-B)^2}, and let G_j, j=1,...,M, be independent standard complex Ginibre matrices of size (nu_{j-1}+n) x (nu_j+n), where nu_0=0, all independent of H; set W_M=G_M^*.... | openai/gpt-6-sol |
2511.10538#0 | 2511.10538 | https://arxiv.org/abs/2511.10538 | math.AP | conjecture | critical_threshold | D | universality | bound_improvement | null | Let S range over all smooth surfaces in R^3 with positive-definite second fundamental form. For g∈L^∞(S,dσ), define E_S g(x)=∫_S e^{ix·ω}g(ω)dσ(ω). What is the critical exponent p_*:=inf{t≥3: for every such S, every p>t, and every ε>0, there exists C_{S,p,ε} such that ||E_S g||_{L^p(B_R^3)}≤C_{S,p,ε}R^ε||g||_{L^∞(S,dσ)... | Give the critical exponent p_*; the original assertion holds exactly when p_*=3. | 3 | true | Let S be a smooth surface in R^3 whose second fundamental form is positive definite, let dσ be its surface measure, and define E_S g(x)=∫_S e^{i x·ω}g(ω)dσ(ω). Is it true that for every p>3, every ε>0, every R>0, and every g∈L^∞(S,dσ), one has ||E_S g||_{L^p(B_R^3)}≤C_{S,p,ε}R^ε||g||_{L^∞(S,dσ)}, where B_R^3 is the bal... | openai/gpt-6-sol |
1705.09251#4 | 1705.09251 | https://arxiv.org/abs/1705.09251 | math.NT | conjecture | critical_threshold | D | universality | bound_improvement | null | What is the infimum of the exponents alpha>0 for which there exists a constant K_alpha>0 depending only on alpha such that, for every elliptic curve E over Q with conductor N_E, Tam(E)<K_alpha N_E^alpha? Here Tam(E)=product_p Tam_p(E), Tam_p(E)=[E(Q_p):E^0(Q_p)], and E^0(Q_p) is the subgroup of points extending to Z_p-... | Report the infimum as a single value; an infimum of 0 means the original assertion holds for every epsilon>0. | 0 | true | For every epsilon>0, does there exist a constant K_epsilon>0 depending only on epsilon such that, for every elliptic curve E over Q with conductor N_E, its global Tamagawa factor Tam(E)=product_p Tam_p(E), where Tam_p(E)=[E(Q_p):E^0(Q_p)] and E^0(Q_p) is the subgroup of points extending to Z_p-sections on the connected... | openai/gpt-6-sol |
2604.13895#0 | 2604.13895 | https://arxiv.org/abs/2604.13895 | math.AP | conjecture | critical_threshold | D | existence | uniqueness | null | Work in R^3. For an open set Omega subset R^3, a function u in H^1_0(Omega) satisfying int_Omega u^2 dx = 1 (with no sign restriction), and q>0, define E_q(u,Omega) = int_Omega |grad u(x)|^2 dx + (q/2) int_Omega int_Omega u(x)u(y)/|x-y| dx dy, and E_q(Omega) = min over all such u of E_q(u,Omega). Among open sets Omega ... | Give the single threshold q_*; use 0 if no such r exists and +infinity if the property holds for every q>0. | null | false | Work in R^3. For an open set Omega subset R^3 and a function u in H^1_0(Omega) with int_Omega u^2 dx = 1 (u not required to have a sign), and a coupling parameter q>0, define E_q(u,Omega) = int_Omega |grad u(x)|^2 dx + (q/2) int_Omega int_Omega u(x)u(y)/|x-y| dx dy (the sum of the Dirichlet energy and a Coulomb/Riesz s... | openai/gpt-6-sol |
2509.14153#0 | 2509.14153 | https://arxiv.org/abs/2509.14153 | math.AP | conjecture | critical_threshold | D | existence | generalization | bound_improvement | For the real-valued Benjamin--Ono initial-value problem ∂_t u=H∂_x²u−2u∂_xu on ℝ_t×ℝ_x, where the Fourier transform of Hf is −i sgn(ξ) times the Fourier transform of f, what is the critical Sobolev index s_*? Global well-posedness holds in H^s(ℝ) for every s>s_*, soliton-based ill-posedness holds for s<s_*, and at s=s_... | Give the single real number s_*; the endpoint s=s_* belongs to the ill-posed regime. | -1/2 | true | For the real-valued Benjamin--Ono initial-value problem \partial_t u=H\partial_x^2u-2u\partial_xu on R_t\times R_x, where \widehat{Hf}(\xi)=-i\,\mathrm{sgn}(\xi)\widehat f(\xi), prove that the problem is ill-posed for initial data in H^{-1/2}(R), in the sense that it does not furnish an evolution with existence, unique... | openai/gpt-6-sol |
1710.08914#0 | 1710.08914 | https://arxiv.org/abs/1710.08914 | math.NT | conjecture | critical_threshold | D | universality | asymptotic_rate | bound_improvement | Let f(u,v)=au^2+buv+cv^2 be a primitive reduced positive definite integral binary quadratic form of discriminant -D=b^2-4ac, where |b|<=a<=c and b>=0 if |b|=a or a=c. Let pi_f(x) count primes p<=x represented by f, let h(-D) be the number of proper equivalence classes of primitive forms of discriminant -D, and set delt... | Give one number; a value of 0 means the stated asymptotic holds throughout x >= (D/a)^(1+epsilon) for every epsilon>0. | 0 | true | Let f(u,v)=au^2+buv+cv^2 be a primitive reduced positive definite integral binary quadratic form of discriminant -D=b^2-4ac, where |b|<=a<=c and b>=0 if |b|=a or a=c. Let pi_f(x) be the number of primes p<=x represented by f, let h(-D) be the number of proper equivalence classes of primitive forms of discriminant -D, a... | openai/gpt-6-sol |
2509.12902#2 | 2509.12902 | https://arxiv.org/abs/2509.12902 | math.NT | conjecture | critical_threshold | D | universality | bound_improvement | null | Let \Gamma be a cocompact torsion-free subgroup of PSL(2,R), and let l be a closed geodesic on \Gamma\backslash\mathbb H. Conjugate so that its axis is the imaginary axis and its associated hyperbolic subgroup is H_1=\langle\operatorname{diag}(m,m^{-1})\rangle for some m>1. For \gamma=\begin{bmatrix}a&b\\c&d\end{bmatri... | Give one real number; the infimum is 0 exactly when the stated estimate holds for every positive epsilon. | 0 | true | Let \Gamma be a cocompact torsion-free subgroup of PSL(2,R), let l be a closed geodesic on \Gamma\backslash\mathbb H, and conjugate so that its axis is the imaginary axis and its associated hyperbolic subgroup is H_1=\langle\mathrm{diag}(m,m^{-1})\rangle for some m>1. For \gamma=\begin{bmatrix}a&b\\c&d\end{bmatrix}\in ... | openai/gpt-6-sol |
2507.22011#13 | 2507.22011 | https://arxiv.org/abs/2507.22011 | math.PR | conjecture | critical_threshold | D | universality | asymptotic_rate | null | For each L in N, in the fixed-q imaginary q-Racah lozenge-tiling model with q=4/5, kappa=3i, and hexagon parameters (T,S,N)=(8L,4L,4L), let b=(b_1,...,b_T), with b_j in {0,1}, be the binary barcode configuration sampled in the bulk of the waterfall region. With the left-boundary offset K=L/10, define h(t)=sum_{j=K}^t b... | The answer is the critical exponent; 0 means the fluctuations grow more slowly than L^ε for every ε>0. | 0 | true | For each L in N, in the fixed-q imaginary q-Racah lozenge-tiling model with q=4/5, kappa=3i, and hexagon parameters (T,S,N)=(8L,4L,4L), let b=(b_1,...,b_T), with b_j in {0,1}, be the binary barcode configuration sampled in the bulk of the waterfall region. With the left-boundary offset K=L/10, define h(t)=sum_{j=K}^t b... | openai/gpt-6-sol |
2503.18864#10 | 2503.18864 | https://arxiv.org/abs/2503.18864 | math.OC | conjecture | critical_threshold | D | universality | generalization | null | Let G be the metric X graph formed by four edges meeting at one central interior vertex, with e1 and e2 of common length ell_b>0 and e3 and e4 of common length ell_t>0. Impose Dirichlet boundary conditions at the four exterior vertices, and let Delta_G act as the second derivative on each edge, with continuity and the ... | Give the single nonnegative real value of T_*; T_*=0 means exact controllability holds for every T>0. | 0 | true | Let G be the metric X graph formed by four edges meeting at one central interior vertex, with e1 and e2 of common length ell_b>0 and e3 and e4 of common length ell_t>0. Impose Dirichlet boundary conditions at the four exterior vertices, and let Delta_G act as the second derivative on each edge, with continuity of funct... | openai/gpt-6-sol |
2602.13737#2 | 2602.13737 | https://arxiv.org/abs/2602.13737 | math.CO | conjecture | critical_threshold | D | universality | generalization | existence_construction | What is the infimum of all η>0 for which there exists n0=n0(η)∈N such that, for every integer n≥n0 divisible by 4, every loopless n-vertex digraph G with at most one edge in each direction between any pair of vertices and δ^0(G)≥(1/2+η)n contains a C4-factor? Here δ^0(G) is the minimum of the indegrees and outdegrees o... | The answer is one number: the stated infimum. | 0 | true | Does the following even-cycle analogue of the asymptotic odd-cycle factor result hold? For every η>0, does there exist n0=n0(η)∈N such that, for every integer n≥n0 divisible by 4, every loopless n-vertex digraph G (with at most one edge in each direction between any pair of vertices) satisfying δ^0(G)≥(1/2+η)n contains... | openai/gpt-6-sol |
2604.00252#0 | 2604.00252 | https://arxiv.org/abs/2604.00252 | math.AP | conjecture | critical_threshold | D | universality | bound_improvement | null | Let T=R/(2pi Z), e_n(x)=(2pi)^(-1/2)e^{inx}, and U(t)=e^{it Delta} on L^2(T), so that U(t)e_n=e^{-itn^2}e_n. Let P_{<=N} project onto Fourier modes |n|<=N. For a finite-rank operator A=sum_j a_j|phi_j><psi_j|, set rho_A(x)=sum_j a_j phi_j(x)overline{psi_j(x)} and rhobar_A(x)=rho_A(x)-(2pi)^(-1)Tr A. What is the infimum... | The answer is the critical shift delta; the original claim holds exactly when this infimum is 0, using the known failure for sigma<2/3-1/alpha. | 0 | true | Let T=R/(2pi Z), let e_n(x)=(2pi)^(-1/2)e^{inx}, let U(t)=e^{it Delta} on L^2(T), so that U(t)e_n=e^{-itn^2}e_n, and let P_{<=N} be the Fourier projection onto {|n|<=N}. For a finite-rank operator A=sum_j a_j|phi_j><psi_j| on L^2(T), define rho_A(x)=sum_j a_j phi_j(x) overline{psi_j(x)} and its renormalised density by ... | openai/gpt-6-sol |
2501.12287#10 | 2501.12287 | https://arxiv.org/abs/2501.12287 | math.CO | conjecture | critical_threshold | D | universality | existence_construction | structural_characterization | What is the critical approximation threshold eta_* = inf{eta>0 : for every R>0 there exist delta_1,delta_2>0 such that the following holds for every finite abelian group G? If f:G->C is 1-bounded and is an (R,delta_1,delta_2)-weak quadratic character—meaning that there is S subseteq G with |S|>=(1-delta_2)|G| such that... | Give the infimum of the approximation tolerances for which the stated uniform approximation property holds. | 0 | true | For every R>0 and eta>0, do there exist delta_1,delta_2>0 such that the following holds for every finite abelian group G? If f:G->C is 1-bounded and is an (R,delta_1,delta_2)-weak quadratic character—meaning that there is S subseteq G with |S|>=(1-delta_2)|G| such that, for every t in S, Delta_t f(x):=overline{f(x)}f(x... | openai/gpt-6-sol |
2503.12471#3 | 2503.12471 | https://arxiv.org/abs/2503.12471 | math.PR | conjecture | critical_threshold | D | extension | bound_improvement | null | For each integer L>=1, let {W(x,.)}_{x=1}^{L-1} be independent two-sided Brownian motions, and let h_* be the almost surely unique minimizer over h:{0,...,L}->R with h(0)=h(L)=0 of E(h)=D(h)-W(h), where D(h)=(1/2)sum_{x=1}^L(h(x)-h(x-1))^2 and W(h)=sum_{x=1}^{L-1}W(x,h(x)). For a random variable X and s>=1, set ||X||_s... | Give the supremum as one number; the value 3 means the strengthened estimate holds for every 1<=s<3. | 3 | true | For every integer L >= 1, let {W(x,.)}_{x=1}^{L-1} be independent two-sided Brownian motions, and let h_* be the almost surely unique minimizer over functions h:{0,...,L}->R with h(0)=h(L)=0 of E(h)=D(h)-W(h), where D(h)=(1/2) sum_{x=1}^L (h(x)-h(x-1))^2 and W(h)=sum_{x=1}^{L-1}W(x,h(x)). For a random variable X and s ... | openai/gpt-6-sol |
2410.01297#3 | 2410.01297 | https://arxiv.org/abs/2410.01297 | math.AP | conjecture | critical_threshold | D | universality | existence_construction | uniqueness | For the standard undamped two-dimensional Boussinesq system on R^2, with no damping term in the velocity equation, near the stable stratified state with zero velocity and scalar profile -x_2, determine the infimum of all ε>0 for which there exist T_ε>0 and a perturbation of this state with H^2(R^2) norm at most ε whose... | The answer is the infimum of the admissible positive ε, with the infimum of the empty set taken as +∞; zero means the original assertion holds. | 0 | true | For the standard undamped two-dimensional Boussinesq system on R^2 (that is, with no damping term in the velocity equation), near the stable stratified state with zero velocity and scalar profile -x_2, does the IPM-type strong H^2 ill-posedness result hold: for every epsilon > 0, can one find T_epsilon > 0 and an H^2(R... | openai/gpt-6-sol |
2411.18637#0 | 2411.18637 | https://arxiv.org/abs/2411.18637 | math.CO | conjecture | critical_threshold | D | universality | bound_improvement | null | What is the supremum of the real numbers Q<1/3 for which the following holds: for every finite graph family F with χ(F)=min{χ(H):H∈F}=3, there is an n₀=n₀(F,Q) such that, for every integer n≥n₀, if ex(n,F)≤e(T_{n,2})+Qn, then every n-vertex F-free graph with maximum adjacency-matrix spectral radius among all n-vertex F... | The answer is this supremum; it equals 1/3 exactly when the original assertion holds for every Q<1/3. | 1/3 | true | For every fixed real Q<1/3 and every finite graph family F with χ(F)=min{χ(H):H in F}=3, is there an n_0=n_0(F,Q) such that, for every integer n>=n_0, if ex(n,F), the maximum number of edges in an n-vertex graph containing no member of F as a subgraph, satisfies ex(n,F)<=e(T_{n,2})+Qn, then every n-vertex F-free graph ... | openai/gpt-6-sol |
2409.00440#0 | 2409.00440 | https://arxiv.org/abs/2409.00440 | math.DG | conjecture | critical_threshold | D | quantity | bound_improvement | existence_construction | Let n be any positive integer, μ>0, g a smooth Riemannian metric on B^n_{1+μ}, and h:(B^n_{1+μ},g)→(R^{n+1},e) a short embedding with h^#e≤g. What is the sharp critical Hölder exponent α_* such that, for every α<α_* and every ε>0, there exists a C^{1,α} isometric embedding f:(B^n_1,g)→(R^{n+1},e) with ||f−h||_0≤ε, wher... | Give one critical exponent; 1/2 corresponds to an affirmative answer to the original question. | 1/2 | true | Let n be a positive integer, mu>0, and let g be a smooth Riemannian metric on B^n_{1+mu}. Given a short embedding h:(B^n_{1+mu},g)->(R^{n+1},e), where h^#e<=g, is alpha=1/2 the sharp regularity threshold for the local codimension-one isometric-embedding h-principle: for every alpha<1/2 and every epsilon>0, does there e... | openai/gpt-6-sol |
2506.03331#7 | 2506.03331 | https://arxiv.org/abs/2506.03331 | math.NT | conjecture | critical_threshold | D | universality | bound_improvement | asymptotic_rate | For p>0 with 2/p∈N, let N_p(r) be the number of points of Z² inside the p-circle {x∈R²: |x₁|^p+|x₂|^p=r^p}, and let P_p(r)=N_p(r)−(2/p)(Γ(1/p)²/Γ(2/p))r². Let q₁^[p] be the decay exponent supplied by the anticipated estimate J_{1,φ}^[p](t)=O(t^(−q₁^[p])) as t→∞, uniform in the distorted angle φ. What is the critical ex... | Give the infimum as one number; ε_*=0 means the proposed bound holds for every admissible p and every sufficiently small ε>0. | 0 | true | Let p>0 satisfy 2/p in N, and for r>0 let N_p(r) be the number of points of Z^2 inside the p-circle {x in R^2 : |x_1|^p+|x_2|^p=r^p}. Define P_p(r)=N_p(r)-(2/p)(Gamma(1/p)^2/Gamma(2/p))r^2. Suppose q_1^{[p]} is the decay exponent supplied by the anticipated asymptotic estimate, uniform with respect to the distorted ang... | openai/gpt-6-sol |
2604.20400#0 | 2604.20400 | https://arxiv.org/abs/2604.20400 | math.NT | conjecture | critical_threshold | D | universality | bound_improvement | null | Let τ(n) be the number of positive divisors of n≥1, and define Δ(x) for real x≥1 by ∑_{n≤x}τ(n)=x(log x+2γ−1)+Δ(x), where γ is the Euler–Mascheroni constant. What is the infimum of the real exponents θ such that, for every ε>0, Δ(x)≪_ε x^{θ+ε} as x tends to infinity? | The answer is the critical exponent for the stated family of bounds. | 1/4 | true | Let tau(n) be the number of positive divisors of the integer n >= 1, and for real x >= 1 define Delta(x) by sum_{n <= x} tau(n) = x(log x + 2gamma - 1) + Delta(x), where gamma is the Euler-Mascheroni constant. Prove that, for every epsilon > 0, Delta(x) <<_epsilon x^{1/4+epsilon} as x tends to infinity. | openai/gpt-6-sol |
2410.06093#1 | 2410.06093 | https://arxiv.org/abs/2410.06093 | math.SP | conjecture | critical_threshold | D | universality | existence_construction | asymptotic_rate | What is the supremum of the values 0<epsilon<1/4 for which there is no explicit constant C(epsilon)>0 such that, for every function g:{n in N:n>=3}->Z_{>=0} satisfying g(n)/n->0 as n->infinity, a Weil-Petersson-random connected finite-area hyperbolic surface X in M_{g(n),n} has, with probability tending to 1 as n->infi... | The answer is the supremum specified in the question, with 0 assigned to the empty set. | 0 | true | For every 0<epsilon<1/4, does there exist an explicit constant C(epsilon)>0 such that, for every function g:{n in N:n>=3}->Z_{>=0} satisfying g(n)/n->0 as n->infinity, a Weil-Petersson-random connected finite-area hyperbolic surface X in the moduli space M_{g(n),n} of genus-g(n) surfaces with n punctures has, with prob... | openai/gpt-6-sol |
2512.17718#1 | 2512.17718 | https://arxiv.org/abs/2512.17718 | math.CO | conjecture | critical_threshold | D | existence | bound_improvement | null | Let R(s,t) be the least n such that every red-blue edge-colouring of K_n contains either a red K_s or a blue K_t. For C>1, let p_C∈(0,1/2) be the unique solution of C=log(p_C)/log(1-p_C). The Gaussian random graph construction samples independent vectors x_1,…,x_n∼N(0,I_d/d), colours ij blue when ⟨x_i,x_j⟩≥-c_p/√d, whe... | Give the supremum as one number, taking it to be 0 if no positive ε satisfies the stated condition. | null | false | Let R(s,t) be the least n such that every red-blue edge-colouring of K_n contains either a red K_s or a blue K_t. For C>1, let p_C∈(0,1/2) be the unique solution of C=log(p_C)/log(1-p_C). The Gaussian random graph construction samples independent vectors x_1,…,x_n∼N(0,I_d/d), colours ij blue when ⟨x_i,x_j⟩≥-c_p/√d, whe... | openai/gpt-6-sol |
2503.12146#0 | 2503.12146 | https://arxiv.org/abs/2503.12146 | math.NT | conjecture | critical_threshold | D | universality | bound_improvement | null | Let D_n(n^{1/2},n^{1/2-ε}) = |{d ∈ N : d divides n and n^{1/2} ≤ d ≤ n^{1/2}+n^{1/2-ε}}|. What is the infimum of the values ε>0 for which there exists a constant k_ε>0 such that D_n(n^{1/2},n^{1/2-ε}) ≤ k_ε for every integer n≥1? | Give one number; 0 means the original assertion holds for every ε>0. | 0 | true | For each fixed epsilon > 0, does there exist a constant k_epsilon > 0 such that, for every integer n >= 1, the number D_n(n^{1/2},n^{1/2-epsilon}) = |{d in N : d divides n and n^{1/2} <= d <= n^{1/2}+n^{1/2-epsilon}}| is at most k_epsilon? | openai/gpt-6-sol |
2509.16980#0 | 2509.16980 | https://arxiv.org/abs/2509.16980 | math.NT | conjecture | critical_threshold | D | universality | bound_improvement | null | What is the infimum of the exponents α such that, for every ε>0, there exists a constant C_{α,ε} for which every odd squarefree positive integer q and every integral ternary quadratic form Q(x_1,x_2,x_3) whose determinant is coprime to q admit a nonzero solution to Q(x_1,x_2,x_3)≡0 (mod q) with max{|x_1|,|x_2|,|x_3|}≤C... | The answer is the critical exponent for the stated uniform bound. | 1/2 | true | For every epsilon > 0, does there exist a constant C_epsilon such that, for every odd squarefree positive integer q and every integral ternary quadratic form Q(x_1,x_2,x_3) whose determinant is coprime to q, the congruence Q(x_1,x_2,x_3) congruent to 0 modulo q has a nonzero solution (x_1,x_2,x_3) in Z^3 with max{|x_1|... | openai/gpt-6-sol |
2507.08819#6 | 2507.08819 | https://arxiv.org/abs/2507.08819 | cond-mat.soft | conjecture | critical_threshold | D | construction | structural_characterization | existence_construction | For the one-dimensional periodic coupled Cahn--Hilliard system r(t)u_t=∂_x²[-ε_u²u_xx-u+u³+αv+βv²] and v_t=∂_x²[-ε_v²v_xx-v+v³+αu+2βuv] on [0,L), where r(t)=τ_u(t)/τ_v(t)>0, consider the paper's linear schedule r(t)=ct+0.01 with c∈[0.1,10] and its fixed initial datum. What is the critical value of c near 0.66 separatin... | Give one numerical value of c at the tipping transition. | null | false | For the one-dimensional periodic coupled Cahn--Hilliard system r(t)u_t = partial_x^2[-epsilon_u^2 u_xx-u+u^3+alpha v+beta v^2] and v_t = partial_x^2[-epsilon_v^2 v_xx-v+v^3+alpha u+2beta uv] on [0,L), where r(t)=tau_u(t)/tau_v(t)>0 is the rescaled ratio of the two relaxation parameters, determine how a prescribed time-... | openai/gpt-6-sol |
1703.06985#0 | 1703.06985 | https://arxiv.org/abs/1703.06985 | math.PR | conjecture | critical_threshold | D | universality | structural_characterization | null | Let (H_N) be a sequence of N × N real symmetric random matrices with bandwidth b_N. Suppose the upper-triangular entries (h_ij)_{i ≤ j} are jointly independent, E h_ij = 0 and E h_ij² = 1 when |i−j| < b_N, and h_ij = 0 when |i−j| ≥ b_N. What is the critical exponent α_c separating the two conjectured universal regimes:... | The answer is the single critical exponent asserted by the original conjecture. | 0.5 | true | Let (H_N) be a sequence of N x N real symmetric random matrices with bandwidth b_N such that the upper-triangular entries (h_ij)_{i <= j} are jointly independent, E h_ij = 0 and E h_ij^2 = 1 when |i-j| < b_N, and h_ij = 0 when |i-j| >= b_N. Prove the Poisson/Gaudin--Mehta conjecture: for every 0 < alpha < 1/2, if b_N i... | openai/gpt-6-sol |
2412.16578#3 | 2412.16578 | https://arxiv.org/abs/2412.16578 | math-ph | conjecture | critical_threshold | D | existence | other | null | For the critical trajectory (separatrix) of x¨+x˙+epsilon x²=0, fix the time-origin by B_0=1 and define B_{n+1}=[(n+1)(n+2)]^{-1} sum_{k=0}^n B_k B_{n-k} for every integer n>=0. With epsilon set to 1 and z=e^{-t}, what is the radius of convergence in z of x(t)=z sum_{n=0}^infinity B_n(-z)^n? | Give the radius of convergence as one number; a positive radius corresponds to convergence as a power series. | null | false | For the critical trajectory (separatrix) of the rescaled equation x¨+x˙+epsilon x^2=0, with epsilon ultimately set to 1, fix the time-origin by B_0=1 and define B_{n+1}=[(n+1)(n+2)]^{-1} sum_{k=0}^n B_k B_{n-k} for every integer n>=0. Does the formal separatrix expansion x(t)=z sum_{n=0}^infinity B_n(-epsilon z)^n, z=e... | openai/gpt-6-sol |
2504.08143#7 | 2504.08143 | https://arxiv.org/abs/2504.08143 | math.AP | conjecture | critical_threshold | D | universality | bound_improvement | structural_characterization | For x>0 and 0<b≤1, define φ_{1,b}(x)=∫_0^{2π} exp(-x√(1+b²-2b cos η))e^{iη} dη, and let φ_1(x)=φ_{1,1}(x). What is the supremum of all t∈[0,1] such that, for every x>0, the map b↦bφ_{1,b}(x) is strictly increasing on (0,t)? | Give the uniform critical threshold as one number; the value 1 means strict increase holds on (0,1) for every x>0. | 1 | true | For each fixed x>0, define φ_{1,b}(x)=∫_0^{2π} exp(-x√(1+b^2-2b cos η))e^{iη} dη for 0<b≤1, so that φ_1(x)=φ_{1,1}(x). Is the map b∈(0,1)↦bφ_{1,b}(x) strictly increasing? Equivalently in its intended consequence, can one prove φ_1(x)-bφ_{1,b}(x)>0 for every b∈(0,1) and every x>0, improving Lemma 5, which establishes th... | openai/gpt-6-sol |
2503.22094#3 | 2503.22094 | https://arxiv.org/abs/2503.22094 | math.CO | conjecture | critical_threshold | D | universality | bound_improvement | asymptotic_rate | Let C_5 be the cycle of length 5. For each integer t >= 2, let r(C_5,t) be the minimum integer n such that every n-vertex graph containing no copy of C_5 has an independent set of size t. It is known that c_1 t^{10/7}/(log t)^{13/7} <= r(C_5,t) <= t^{3/2}/(log t)^{1/2} for every integer t >= 3 and some constant c_1 > 0... | The answer is the critical exponent; it equals 3/2 precisely when r(C_5,t) = t^{3/2-o(1)} as t tends to infinity. | 3/2 | true | Let C_5 be the cycle of length 5, and, for each integer t >= 2, let r(C_5,t) be the minimum integer n such that every n-vertex graph containing no copy of C_5 has an independent set of size t. The known bounds are c_1 t^{10/7}/(log t)^{13/7} <= r(C_5,t) <= t^{3/2}/(log t)^{1/2} for all t >= 3 and some constant c_1 > 0.... | openai/gpt-6-sol |
1801.06515#6 | 1801.06515 | https://arxiv.org/abs/1801.06515 | math.FA | conjecture | critical_threshold | D | universality | structural_characterization | null | For every 0<p≤2, let 𝓗^p be the completion of Dirichlet polynomials f(s)=∑_{n=1}^N a_n n^{-s} in the norm (quasi-norm if p<1) ‖f‖_{𝓗^p}=(lim_{T→∞}(2T)^{-1}∫_{-T}^T |f(it)|^p dt)^{1/p}, and set φ_{1/p}(s)=1+∑_{n=2}^∞[n^{1/2}(log n)^{1/p}]^{-1}n^{-s}. What is the critical value p_* such that, for 0<p≤2, there exists C_... | Give the single critical value p_*. | 1 | true | For every 0<p\leq2, let \mathcal{H}^p be the completion of Dirichlet polynomials f(s)=\sum_{n=1}^N a_n n^{-s} in the norm (quasi-norm if p<1) \|f\|_{\mathcal{H}^p}=(\lim_{T\to\infty}(2T)^{-1}\int_{-T}^T|f(it)|^p\,dt)^{1/p}, and set \varphi_{1/p}(s)=1+\sum_{n=2}^\infty[n^{1/2}(\log n)^{1/p}]^{-1}n^{-s}. Is it true that ... | openai/gpt-6-sol |
1708.03742#0 | 1708.03742 | https://arxiv.org/abs/1708.03742 | math.NT | conjecture | critical_threshold | D | universality | bound_improvement | null | Let ε* be the infimum of all ε>0 for which there exists c_ε>0 such that every finite set A={a_1<...<a_k}⊂R satisfying a_i−a_{i−1}<a_{i+1}−a_i for every integer i with 1<i<k obeys both |A+A|≥c_ε k^{2−ε} and |A−A|≥c_ε k^{2−ε}, where A+A={a+a':a,a'∈A} and A−A={a−a':a,a'∈A}. What is ε*? | The answer is the critical infimum ε*; a value of 0 means the original assertion holds for every ε>0. | 0 | true | For every epsilon>0, does there exist a constant c_epsilon>0 such that every finite set A={a_1<...<a_k} subset R whose consecutive gaps are strictly increasing, i.e. a_i-a_{i-1}<a_{i+1}-a_i for every integer i with 1<i<k, satisfies both |A+A|>=c_epsilon k^{2-epsilon} and |A-A|>=c_epsilon k^{2-epsilon}, where A+A={a+a':... | openai/gpt-6-sol |
2503.18009#4 | 2503.18009 | https://arxiv.org/abs/2503.18009 | math.NT | conjecture | critical_threshold | D | universality | generalization | null | For every natural number r, every j∈Z with (j,r)=1, and every integer R with 1≤R≤r, let E_2(R;j,r) count quadruples (k_1,k_2,k_3,k_4) modulo r such that k_1+k_2≡k_3+k_4 (mod r) and, for each i, k_i²≡jm_i (mod r) for some integer 1≤m_i≤R. Let E_4(R;j,r) analogously count octuples (k_1,…,k_8) modulo r such that k_1+⋯+k_4... | Give the infimum as one number; 0 means the original estimates hold for every ε>0. | 0 | true | For every epsilon>0, every natural number r, every j in Z with (j,r)=1, and every integer R with 1 <= R <= r, let E_2(R;j,r) be the number of quadruples (k_1,k_2,k_3,k_4) modulo r such that k_1+k_2 congruent to k_3+k_4 modulo r and, for each i, k_i^2 congruent to jm_i modulo r for some integer m_i with 1 <= m_i <= R; l... | openai/gpt-6-sol |
1709.01146#7 | 1709.01146 | https://arxiv.org/abs/1709.01146 | math.NT | conjecture | critical_threshold | D | universality | asymptotic_rate | null | Start with the ordered sequence 1,2,3,... . At stage n≥1, let x_n be the least element other than 1 remaining from the preceding stage, and form the next stage by deleting every x_n-th element of that preceding remaining sequence, including x_n itself (the hard counting sieve). For n≥1, let d_n=∏_{i=1}^n(1−1/x_i), the ... | Give the infimum as a single number; use +∞ if no ε>0 satisfies the estimate. | 0 | true | Start with the ordered sequence 1,2,3,... . At stage n>=1, let x_n be the least element other than 1 remaining from the preceding stage, and form the next stage by deleting every x_n-th element of that preceding remaining sequence, including x_n itself (the hard counting sieve). For n>=1 let d_n=product_{i=1}^n(1-1/x_i... | openai/gpt-6-sol |
1710.02737#3 | 1710.02737 | https://arxiv.org/abs/1710.02737 | math.AP | conjecture | critical_threshold | D | construction | structural_characterization | generalization | For the De Gregorio equation on S^1, omega_t + u omega_theta = u_theta omega, with u_theta = H omega and integral_{S^1} u dtheta = 0, where H is the periodic Hilbert transform, let M_m = {A sin(m(theta-theta_0)) : A in R, theta_0 in (-pi,pi]} for each integer m >= 1. What is the critical Sobolev exponent s_* for a theo... | Give the single critical Sobolev exponent s_*. | 3/2 | true | For the De Gregorio equation on S^1, omega_t + u omega_theta = u_theta omega, with u_theta = H omega and integral_{S^1} u dtheta = 0, where H is the periodic Hilbert transform, let M_m = {A sin(m(theta-theta_0)) : A in R, theta_0 in (-pi,pi]} for each integer m >= 1. Can one formulate and prove a precise dynamical-inst... | openai/gpt-6-sol |
2506.15353#3 | 2506.15353 | https://arxiv.org/abs/2506.15353 | math-ph | conjecture | critical_threshold | D | universality | asymptotic_rate | generalization | For each N, let \(\mathcal Q_N=\{-1,1\}^N\), \(\mathcal H=\ell^2(\mathcal Q_N)\), and let \(U(\sigma)\), \(\sigma\in\mathcal Q_N\), be independent centered Gaussian random variables of variance N. Define \((Tf)(\sigma)=\sum_{\tau:d(\sigma,\tau)=1}f(\tau)\), \(|-\rangle=2^{-N/2}\sum_{\sigma\in\mathcal Q_N}|\sigma\rangle... | Give the single exact value of the critical threshold b. | sqrt(2 ln 2) | true | Let \(\mathcal Q_N=\{-1,1\}^N\), let \(\mathcal H=\ell^2(\mathcal Q_N)\), and let \(U(\sigma)\), \(\sigma\in\mathcal Q_N\), be independent centered Gaussian random variables of variance \(N\). Write \(T\) for the Hamming-cube adjacency operator, \((Tf)(\sigma)=\sum_{\tau:d(\sigma,\tau)=1}f(\tau)\), and \(|-\rangle=2^{-... | openai/gpt-6-sol |
2502.00143#7 | 2502.00143 | https://arxiv.org/abs/2502.00143 | math.SP | conjecture | critical_threshold | D | universality | bound_improvement | asymptotic_rate | Let S be a compact smooth simple symmetric surface of revolution in mathfrak{S}. In meridian–angular coordinates its metric is f(nu)^2 dtheta^2+dnu^2; f has one nondegenerate maximum (the equator) and is symmetric about that maximum. Its normalized geodesic phase shift omega(I), I in [0,1], satisfies either omega'(I)<0... | The answer is the critical exponent tau_*, if one exists with both stated properties. | 1/3 | true | Let S be a compact smooth simple symmetric surface of revolution in mathfrak{S}: in meridian--angular coordinates its metric is f(nu)^2 dtheta^2+dnu^2, f has one nondegenerate maximum (the equator), is symmetric about that maximum, and its normalized geodesic phase shift omega(I), I in [0,1], satisfies either omega'(I)... | openai/gpt-6-sol |
1705.06581#0 | 1705.06581 | https://arxiv.org/abs/1705.06581 | math.CO | conjecture | critical_threshold | D | existence | bound_improvement | null | What is the infimum of all real ε>1/6 for which there exists an absolute constant C>0 such that, for every prime power q and every set A⊆𝔽_q, |A|>Cq^(1/2+ε) implies (A−A)(A−A)=𝔽_q, where (A−A)(A−A)={(a−b)(c−d):a,b,c,d∈A}? The known sufficient exponent is 3/4. | Give the infimum as one number. | null | false | Does there exist an absolute epsilon>1/6 and an absolute constant C>0 such that, for every prime power q and every set A subset of the finite field F_q, |A|>Cq^{1/2+epsilon} implies (A-A)(A-A)=F_q, where (A-A)(A-A)={ (a-b)(c-d) : a,b,c,d in A }? Equivalently in threshold terms, can the known sufficient exponent 3/4 for... | openai/gpt-6-sol |
2409.06079#0 | 2409.06079 | https://arxiv.org/abs/2409.06079 | math.PR | conjecture | critical_threshold | D | universality | bound_improvement | asymptotic_rate | Fix q≥2 and sufficiently large β. In the q-state Potts model μ_{Λ_n}^{fl} on the n×n×n box Λ_n, impose blue boundary conditions on the bottom face (at height 0) and red boundary conditions on the other five faces. Let I_Blue be the plaquette interface separating the blue phase, including the component connected to the ... | Give the critical value for the fixed q and β; a value of 0 means the bound holds for every δ>0. | 0 | true | Fix q>=2 and sufficiently large beta. In the q-state Potts model mu_{Lambda_n}^{fl} on the n x n x n box Lambda_n, impose blue boundary conditions on its bottom face (at height 0) and red boundary conditions on its other five faces. Let I_Blue be the plaquette interface separating the blue phase, including the componen... | openai/gpt-6-sol |
2409.08819#20 | 2409.08819 | https://arxiv.org/abs/2409.08819 | math.CO | conjecture | critical_threshold | D | universality | bound_improvement | null | Let Q_k be the Boolean lattice of all subsets of a k-element set, ordered by inclusion. Let R(P,Q) be the least N in N such that every blue/red coloring of the vertices of Q_N contains either a blue induced copy of P or a red induced copy of Q. The established bound for 2^25 <= m <= n is R(Q_m,Q_n) <= n(m-(1-2/sqrt(log... | Give one real number; epsilon_* = 0 means the bound holds for every epsilon > 0. | 0 | true | Let Q_k be the Boolean lattice consisting of all subsets of a k-element set, ordered by inclusion, and let R(P,Q) be the least N in N such that every blue/red coloring of the vertices of Q_N contains either a blue induced copy of P or a red induced copy of Q. The established bound for 2^25 <= m <= n is R(Q_m,Q_n) <= n(... | openai/gpt-6-sol |
2012.15809#0 | 2012.15809 | https://arxiv.org/abs/2012.15809 | math.NT | conjecture | critical_threshold | D | universality | bound_improvement | asymptotic_rate | Let f be either of two random multiplicative functions. In the Steinhaus model, the variables f(p), for primes p, are independent and uniform on {z in C: |z|=1}, and f(n)=product_{p^a || n} f(p)^a for every n>=1. In the Rademacher model, the variables f(p) are independent and each equal to +1 or -1 with probability 1/2... | Give the infimum as one extended-real number, taking it to be +infinity if no epsilon qualifies. | 0 | true | Let f be either of the following random multiplicative functions. In the Steinhaus model, the variables f(p), for primes p, are independent and uniform on {z in C: |z|=1}, and f(n)=product_{p^a || n} f(p)^a for every n>=1. In the Rademacher model, the variables f(p) are independent and each equal to +1 or -1 with proba... | openai/gpt-6-sol |
2503.13867#2 | 2503.13867 | https://arxiv.org/abs/2503.13867 | math.AP | conjecture | critical_threshold | D | quantity | bound_improvement | null | In the codimension-one case of surface immersions into R^3, let alpha_0 be the supremum of exponents alpha in (0,1) for which the following holds for every smooth bounded simply connected domain Omega in R^2 and every C^2 Riemannian metric g on its closure: every C^1 immersion u from the closure of Omega into R^3 satis... | Give one number; 1/3 means that the known flexibility range alpha < 1/3 is sharp for the stated approximation property. | 1/3 | true | Consider isometric immersions of surfaces into three-dimensional Euclidean space, i.e. the codimension-one case n=2. Fix a smooth bounded simply connected domain Omega in R^2 and a C^2 Riemannian metric g on the closure of Omega (a C^2 field of symmetric positive-definite 2x2 matrices). Call an immersion u in C^1(closu... | openai/gpt-6-sol |
2501.09470#8 | 2501.09470 | https://arxiv.org/abs/2501.09470 | math.NT | conjecture | critical_threshold | D | universality | bound_improvement | null | Let α* be the supremum of all t≥0 such that, for every ε>0, there is a constant c_{t,ε}>0 for which both |A+A|≥c_{t,ε}κ^{-t+ε}|A| and |A-A|≥c_{t,ε}κ^{-t+ε}|A| hold for every abelian group G and finite A⊆G with κ∈(0,1] minimal such that, for every finite B⊆G, ∑_{x∈G}(1_A*1_B)(x)^3≤κ|A|²|B|². Here 1_S is the indicator of... | Give one number; α*=1 is equivalent to a Yes answer to the original question. | 1 | true | Let G be an arbitrary abelian group, let A\subseteq G be finite, and let \kappa\in(0,1] be minimal such that, for every finite B\subseteq G, \sum_{x\in G}(1_A*1_B)(x)^3\leq\kappa|A|^2|B|^2, where 1_S denotes the indicator of S and (f*g)(x)=\sum_{y\in G}f(x-y)g(y). The known bounds |A+A|\gg_\epsilon\kappa^{-11/19+\epsil... | openai/gpt-6-sol |
2504.04208#1 | 2504.04208 | https://arxiv.org/abs/2504.04208 | math.CO | conjecture | critical_threshold | D | universality | hypothesis_weakening | null | What is the infimum of the values c>0 for which there exist C=C(c)>0, an integer d=d(c)>=1, and n_0=n_0(c) such that, for every n>=n_0 and every set P subset R^2 with |P|=n that determines at least c n^{4/3} pairs (p,p') in P x P satisfying ||p-p'||=1, there is a subset P' subset P with |P'|>=n/C contained in a general... | Give the infimum as a single number; 0 means the stated conclusion holds for every c>0. | 0 | true | For every c>0, do there exist constants C=C(c)>0, an integer d=d(c)>=1, and n_0=n_0(c) such that, for every n>=n_0 and every set P subset R^2 with |P|=n that determines at least c n^{4/3} pairs (p,p') in P x P with ||p-p'||=1, there is a subset P' subset P with |P'|>=n/C contained in a generalized arithmetic progressio... | openai/gpt-6-sol |
1709.03282#1 | 1709.03282 | https://arxiv.org/abs/1709.03282 | math.NT | conjecture | critical_threshold | D | universality | bound_improvement | asymptotic_rate | Let Gamma be a finite-volume Fuchsian subgroup of PSL(2,R) acting on the upper half-plane H, and let z,w lie in H. For X>2, set u(z,w)=|z-w|^2/(4 Im(z) Im(w)) and N(z,w,X)=#{gamma in Gamma: 4u(gamma z,w)+2 <= X}. Let {u_j}_{j>=0} be a complete orthonormal system of Maass forms for Gamma with Delta u_j=lambda_j u_j, lam... | Give one real number; the original estimate holds for every epsilon>0 exactly when this infimum is 1/2. | 1/2 | true | Let Gamma be a finite-volume Fuchsian subgroup of PSL(2,R) acting on the upper half-plane H, and fix z,w in H. For X>2, set u(z,w)=|z-w|^2/(4 Im(z) Im(w)) and N(z,w,X)=#{gamma in Gamma: 4u(gamma z,w)+2 <= X}. Let {u_j}_{j>=0} be a complete orthonormal system of Maass forms for Gamma with Delta u_j=lambda_j u_j, lambda_... | openai/gpt-6-sol |
2412.11062#15 | 2412.11062 | https://arxiv.org/abs/2412.11062 | math.CA | conjecture | critical_threshold | D | existence | existence_construction | null | Let p_* be the supremum of all p∈[0,1) such that, for every increasing sequence (a_n)_{n=1}^∞ of real numbers with a_n→∞, there exists a measurable set E⊆R satisfying m(E∩(k+[0,1]))≥p for every k∈Z and (x+ya_n)_{n=1}^∞ is not contained in E for every x∈R and every y∈R\{0}. What is p_*? | Give one real number; p_*=1 means the stated set exists for every permitted sequence and every p∈[0,1). | 1 | true | Let (a_n)_{n=1}^∞ be any increasing sequence of real numbers with a_n → ∞, and let 0 ≤ p < 1. Does there exist a measurable set E ⊆ R such that m(E ∩ (k+[0,1])) ≥ p for every k ∈ Z and such that, for every x ∈ R and every y ∈ R\{0}, the affine copy (x+y a_n)_{n=1}^∞ is not contained in E? Equivalently, is every such se... | openai/gpt-6-sol |
1704.05870#1 | 1704.05870 | https://arxiv.org/abs/1704.05870 | math.PR | conjecture | critical_threshold | D | existence | bound_improvement | null | Let (X_n)_{n≥0} be a d-dimensional simple random walk on Z^d started at 0. The baseline bound states that, for each integer d≥4, the probability that its trace covers any nearest-neighbor path from 0 to the L1-boundary at radius N is at most P_d^{⌊N/d⌋} for some P_d∈(0,1). What is the supremum of {0} together with all ... | The answer is the supremum of the admissible positive exponents, taken to be 0 if there are none. | null | false | For the baseline result that, for each d≥4, the probability that a d-dimensional simple random walk (X_n)_{n≥0} on Z^d started at 0 covers the trace of any nearest-neighbor path from 0 to the L1-boundary at radius N is at most P_d^{⌊N/d⌋} for some P_d∈(0,1), does there exist ε>0 such that, for every integer d≥4, there ... | openai/gpt-6-sol |
2511.10641#3 | 2511.10641 | https://arxiv.org/abs/2511.10641 | math.CO | conjecture | critical_threshold | D | existence | bound_improvement | asymptotic_rate | Let C_4 be the 4-cycle, K_k the complete graph on k vertices, and r(C_4,K_k) the least n such that every graph on n vertices contains either a copy of C_4 or an independent set of size k. What is the supremum of all epsilon > 0 for which r(C_4,K_k) <= k^{2-epsilon+o(1)} as k -> infinity, where o(1) tends to 0 as k -> i... | The answer is the critical exponent epsilon_*; the original question has answer Yes exactly when epsilon_* > 0. | null | false | Does there exist a constant epsilon > 0 such that, as k -> infinity, r(C_4,K_k) <= k^{2-epsilon+o(1)}, where C_4 is the 4-cycle, K_k is the complete graph on k vertices, r(C_4,K_k) is the least n such that every graph on n vertices contains either a copy of C_4 or an independent set of size k, and o(1) denotes a quanti... | openai/gpt-6-sol |
2511.08501#2 | 2511.08501 | https://arxiv.org/abs/2511.08501 | math.CO | conjecture | critical_threshold | D | extension | bound_improvement | asymptotic_rate | What is the supremum of the real numbers c>0 for which the authors’ methods can be refined to prove the following: for every sufficiently large integer k, every integer r with 2 <= r <= k, and every k-uniform hypergraph H with m edges, H has an r-cut whose surplus over the random r-cut baseline S(k,r)r!m/r^k is Omega(m... | Give the supremum, taking it to be 0 if no positive c is admissible and +∞ if the admissible values are unbounded. | null | false | Can the authors' methods be refined to prove that there is a constant c>0 such that, for every sufficiently large integer k, every integer r with 2 <= r <= k, and every k-uniform hypergraph H with m edges, H has an r-cut (a partition of V(H) into r parts, counting an edge when it meets every part) whose surplus over th... | openai/gpt-6-sol |
2602.12928#0 | 2602.12928 | https://arxiv.org/abs/2602.12928 | math.CO | conjecture | critical_threshold | D | characterization | asymptotic_rate | null | For each fixed λ>0 and α>0, start with the ordered deck {1,…,n}. In one asymmetric single-shelf shuffle, draw cards from the bottom and independently place each on top with probability p_n=1−λ/n^α and on the bottom with probability 1−p_n. Let X_n be the number of correct guesses under the optimal full-feedback strategy... | Give the single numerical value of α_c. | null | false | For each fixed lambda>0 and alpha>0, start with the ordered deck {1,...,n}, perform one asymmetric single-shelf shuffle in which cards are drawn from the bottom and independently placed on top with probability p_n=1-lambda/n^alpha and on the bottom with probability 1-p_n, and let X_n be the number of correct guesses ma... | openai/gpt-6-sol |
1706.05642#4 | 1706.05642 | https://arxiv.org/abs/1706.05642 | math.CO | conjecture | critical_threshold | D | existence | structural_characterization | existence_construction | What is the supremum of the set consisting of 0 and all ε>0 for which there exists n₀ such that, for every integer n≥n₀ with 5∣n and every n-vertex graph G satisfying δ(G)≥(1−ε)n, every triangle-free subgraph F⊆G that maximizes the number of copies of C₅ among all triangle-free subgraphs of G admits a partition V(G)=V₀... | The answer is this supremum; the original question has answer Yes exactly when it is positive. | null | false | Do there exist constants \(\varepsilon>0\) and \(n_0\) such that, for every integer \(n\ge n_0\) with \(5\mid n\) and every graph \(G\) on \(n\) vertices satisfying \(\delta(G)\ge(1-\varepsilon)n\), every triangle-free subgraph \(F\subseteq G\) that maximizes the number of copies of \(C_5\) among all triangle-free subg... | openai/gpt-6-sol |
2501.07644#9 | 2501.07644 | https://arxiv.org/abs/2501.07644 | math.CO | conjecture | critical_threshold | D | existence | bound_improvement | null | For each positive integer n, let T_n be the largest integer B in {0,...,n} such that every edge-colouring chi:E(K_{n,n})->N with |chi^{-1}(c)|<=B for every colour c contains a rainbow perfect matching. What is the critical asymptotic value liminf_{n->infinity} T_n/n? | Give one real number; the value is 1 exactly when there exists a function f with f(n)=o(n) such that, for all sufficiently large n, every globally (n-f(n))-bounded colouring contains a rainbow perfect matching. | 1 | true | Consider arbitrary edge-colourings chi : E(K_{n,n}) -> N of the balanced complete bipartite graph K_{n,n} with n vertices in each part; the colourings are not assumed to be proper. For a real b > 0, call chi globally b-bounded if every colour class has size at most b, that is |chi^{-1}(c)| <= b for every colour c. Call... | openai/gpt-6-sol |
2512.13849#2 | 2512.13849 | https://arxiv.org/abs/2512.13849 | math.CO | conjecture | critical_threshold | D | universality | bound_improvement | null | Let A={a_1<...<a_n} be a finite subset of R with strictly increasing consecutive gaps a_{j+1}-a_j. Write A+A={a+b:a,b in A} and A-A={a-b:a,b in A}. What is the infimum of the values epsilon>0 for which there exists a constant c>0, independent of A, such that min(|A+A|,|A-A|) >= c|A|^(2-epsilon) for every such A? | The answer is the infimum over epsilon>0; it need not be attained. | 0 | true | Let A={a_1<a_2<...<a_n} be a finite subset of R whose consecutive gaps a_{j+1}-a_j are strictly increasing, and write A+A={a+b:a,b in A} and A-A={a-b:a,b in A}. For every epsilon>0, does there exist a constant c>0, independent of A, such that min(|A+A|,|A-A|) >= c|A|^(2-epsilon)? This would strengthen the separately es... | openai/gpt-6-sol |
2510.26134#2 | 2510.26134 | https://arxiv.org/abs/2510.26134 | math.CO | conjecture | critical_threshold | D | universality | asymptotic_rate | bound_improvement | What is the critical value \(\varepsilon_* = \inf\{\varepsilon>0: \text{there exists an integer }M\text{ such that every finite poset }P\text{ with }\pi(P)>M\text{ satisfies }\delta(P)>1/2-\varepsilon\}\)? Here \(\pi(P)=\max_{x\in P}|\{y\in P:y\not\sim x\}|\), \(x\sim y\) means \(x<y\) or \(x>y\) in \(P\), and \(\delta... | Give the infimum as one number; the original statement holds exactly when that number is 0. | 0 | true | For every \varepsilon>0, does there exist an integer M such that every finite poset P with \pi(P)>M satisfies \delta(P)>1/2-\varepsilon, where \pi(P)=\max_{x\in P}|\{y\in P:y\not\sim x\}|, x\sim y means that x<y or x>y in P, and \delta(P)=\max_{x\ne y}\min\{\Pr(x\prec y),\Pr(y\prec x)\}, with \prec a uniformly random l... | openai/gpt-6-sol |
2409.08819#26 | 2409.08819 | https://arxiv.org/abs/2409.08819 | math.CO | conjecture | critical_threshold | D | quantity | asymptotic_rate | bound_improvement | For each n in N, let Q_n be the poset of all subsets of an n-element ground set ordered by inclusion, and let R(Q_n,Q_n) be the least N in N such that every blue/red coloring of the vertices of Q_N contains a blue induced copy of Q_n or a red induced copy of Q_n. Given 2.02n+o(1) <= R(Q_n,Q_n) <= n^2-(1-o(1))n log n as... | The answer is one real number; it equals 1 precisely when R(Q_n,Q_n)=O(n^(1+o(1))). | 1 | true | For each n in N, let Q_n be the poset of all subsets of an n-element ground set ordered by inclusion, and let R(Q_n,Q_n) be the least N in N such that every blue/red coloring of the vertices of Q_N contains a blue induced copy of Q_n or a red induced copy of Q_n. Given the bounds 2.02n+o(1) <= R(Q_n,Q_n) <= n^2-(1-o(1)... | openai/gpt-6-sol |
2510.24691#0 | 2510.24691 | https://arxiv.org/abs/2510.24691 | math.CO | conjecture | critical_threshold | D | universality | hypothesis_weakening | null | For positive integers k and ell, let N(n,k,ell) be the maximum, over all n-vertex graphs G, of the number of k-vertex subsets X of V(G) for which G[X] has exactly ell edges, and let ind(k,ell) = lim_{n -> infinity} N(n,k,ell)/binom(n,k). What is the infimum of the real numbers epsilon > 0 for which there exists an inte... | The answer is the infimum over the specified positive values of epsilon; it is 0 exactly when the original assertion holds. | 0 | true | For positive integers k and ell, let N(n,k,ell) be the maximum, over all n-vertex graphs G, of the number of k-vertex subsets X of V(G) for which the induced graph G[X] has exactly ell edges, and let ind(k,ell) = lim_{n -> infinity} N(n,k,ell)/binom(n,k). Is it true that, for every epsilon > 0, whenever k is sufficient... | openai/gpt-6-sol |
2509.00586#0 | 2509.00586 | https://arxiv.org/abs/2509.00586 | math.CO | conjecture | critical_threshold | D | existence | bound_improvement | null | Let ε* be the supremum of {0} together with all real ε₁>0 such that, for every odd prime p and all positive integers k,c,d satisfying 36 log p ≤ d ≤ 3 log c, every k-by-c 0-1 matrix M over F_p with pairwise distinct rows, pairwise distinct columns, and rank_F_p(M)=d satisfies d ≥ (1+ε₁)(log k−2). What is ε*? This is th... | Report the single value ε*; the original existence question has a positive answer exactly when ε*>0. | null | false | Does there exist an absolute constant epsilon_1>0 such that, for every odd prime p and all positive integers k,c,d with 36 log p <= d <= 3 log c, every k-by-c 0-1 matrix M over F_p having pairwise distinct rows and pairwise distinct columns and rank_F_p(M)=d satisfies d >= (1+epsilon_1)(log k-2)? Equivalently in scope,... | openai/gpt-6-sol |
2408.12835#3 | 2408.12835 | https://arxiv.org/abs/2408.12835 | math.CO | conjecture | critical_threshold | D | existence | existence_construction | null | What is the critical value C_* = inf{C>1 : the following property holds}? As n tends to infinity, for every graph G=(V,E) with n=|V| and every family S=(S_v:v∈V) satisfying |S_v|=d_G(v)+1 for every v, whenever C log n is an admissible subset size, choose L(v) independently and uniformly from the C log n-element subsets... | Give the infimum as one number, taking it to be +∞ if no C>1 satisfies the property. | null | false | Does there exist an absolute constant C_9>1 such that, as n -> infinity, the following holds for every graph G=(V,E) with n=|V| vertices and every family S=(S_v:v in V) satisfying |S_v|=d_G(v)+1 for every v (whenever C_9 log n is an admissible subset size): if, independently for each v in V, L(v) is chosen uniformly fr... | openai/gpt-6-sol |
2603.29626#0 | 2603.29626 | https://arxiv.org/abs/2603.29626 | math.CO | conjecture | critical_threshold | D | universality | bound_improvement | existence_construction | Let ε_* be the infimum of the real numbers ε>0 for which every orientation G (that is, every directed graph with no directed 2-cycle) contains a vertex v satisfying |N_2(G,v)| ≥ (1−ε)|N_1(G,v)|, where N_1(G,v)={w∈V(G):(v,w)∈E(G)} and N_2(G,v)={w∈V(G)\(N_1(G,v)∪{v}): there exists u∈N_1(G,v) with (u,w)∈E(G)}. What is ε_*... | The answer is the infimum of the positive ε for which the stated property holds for every orientation. | 0 | true | For every real epsilon > 0, does every orientation G (that is, every directed graph with no directed 2-cycle) contain a vertex v such that |N_2(G,v)| >= (1-epsilon)|N_1(G,v)|, where N_1(G,v)={w in V(G):(v,w) in E(G)} and N_2(G,v)={w in V(G)\(N_1(G,v) union {v}): there exists u in N_1(G,v) with (u,w) in E(G)}? | openai/gpt-6-sol |
1710.11281#4 | 1710.11281 | https://arxiv.org/abs/1710.11281 | math.CO | conjecture | critical_threshold | D | universality | asymptotic_rate | null | For every graph G, let c(G) be its cop number, g(G) its orientable genus, and \widetilde g(G) its nonorientable genus (crosscap number). For each g, define c(g)=max{c(G):g(G)<=g} and \widetilde c(g)=max{c(G):\widetilde g(G)<=g}. What is the infimum of the numbers epsilon>0 for which there exists g0 such that, for every... | The answer is the critical epsilon threshold, whose value is 0 exactly when the original assertion holds. | 0 | true | For every graph G, let c(G) be its cop number, let g(G) be its orientable genus, and let \widetilde g(G) be its nonorientable genus (crosscap number). For each g, define c(g)=max{c(G): g(G)<=g} and \widetilde c(g)=max{c(G): \widetilde g(G)<=g}. Is it true that, for every epsilon>0, there exists g0 such that for every g... | openai/gpt-6-sol |
2412.19756#3 | 2412.19756 | https://arxiv.org/abs/2412.19756 | math.CO | conjecture | critical_threshold | D | extension | generalization | existence_construction | What is the supremum of the constants epsilon in (0,1) for which the following assertion fails: there exists an integer d0 such that, for every d>=d0, every n-vertex (n,d,lambda)-graph G in a family with lambda=o(d), and every tree T with |V(T)| dividing n and |V(T)|<=(1-epsilon)d/log d, G contains a T-factor—that is, ... | The answer is this supremum, with 0 meaning that the assertion holds for every epsilon in (0,1). | 0 | true | For every constant epsilon with 0<epsilon<1, does there exist an integer d0 such that, for every d>=d0, every n-vertex (n,d,lambda)-graph G in a family with lambda=o(d), and every tree T with |V(T)| dividing n and |V(T)|<=(1-epsilon)d/log d, G contains a T-factor, i.e. n/|V(T)| vertex-disjoint copies of T whose union i... | openai/gpt-6-sol |
2505.07756#4 | 2505.07756 | https://arxiv.org/abs/2505.07756 | math.CO | conjecture | critical_threshold | D | construction | bound_improvement | existence_construction | For integers d>=2, let [2d+2]={1,...,2d+2}. What is the supremum of the constants c for which one can give a systematic construction of families F_d subseteq binom([2d+2],d+1) such that no (d+1)-element subset of [2d+2] is shattered by F_d and liminf_{d->infinity} |F_d|/binom(2d+2,d+1) >= c? | The answer is the supremum asymptotic density; a value greater than 9/16 means the coefficient in the original question can be increased by a positive constant. | null | false | For integers d>=2, let [2d+2]={1,...,2d+2}. Can one give a systematic construction of families F_d subseteq binom([2d+2],d+1) with VC-dimension at most d (that is, no (d+1)-element subset of [2d+2] is shattered by F_d) whose size asymptotically improves on the Ahlswede-Khachatrian lower bound binom(2d+1,d)+binom(2d-2,d... | openai/gpt-6-sol |
2412.01121#14 | 2412.01121 | https://arxiv.org/abs/2412.01121 | math.CO | conjecture | critical_threshold | D | universality | bound_improvement | generalization | A k-graph H=(V,E) has edges that are k-element subsets of V. For S⊆V, let deg_H(S) be the number of edges containing S, and for d≥0 let δ_d(H)=min_{S⊆V, |S|=d} deg_H(S). A k-graph system of order n consists of n, not necessarily distinct, k-graphs H_1,...,H_n on a common n-vertex set; its minimum d-degree is min_{i∈[n]... | The answer is the infimum over μ>0 of the values for which both stated implications hold for every admissible k,d and all sufficiently large n; 0 means they hold for every μ>0. | 0 | true | Fix notation. A k-graph (k-uniform hypergraph) H=(V,E) has edge set E a family of k-element subsets of V. For a set S subset of V, deg_H(S) is the number of edges of H containing S, and for an integer d>=0 the minimum d-degree is delta_d(H)=min{deg_H(S): S subset of V with |S|=d}. A k-graph system of order n is a colle... | openai/gpt-6-sol |
2502.00176#8 | 2502.00176 | https://arxiv.org/abs/2502.00176 | math.CO | conjecture | critical_threshold | D | existence | bound_improvement | generalization | What is the infimum of the real exponents c for which the following holds: for every sequence n -> infinity and every p=p(n) in [0,1] with pn an integer and p >> log^c n/n, if R(n,n,p) is chosen uniformly from the pn-regular bipartite graphs with two vertex classes of size n, then for some p'=p'(n) with p' ~ p there is... | Give the infimum as one extended-real value, taking it to be +∞ if no exponent works. | null | false | Does there exist an absolute constant c such that, for every sequence n -> infinity and every p=p(n) in [0,1] with pn an integer and p >> log^c n/n, if R(n,n,p) is chosen uniformly from the pn-regular bipartite graphs with two vertex classes of size n, then for some p'=p'(n) with p' ~ p there is a joint distribution of... | openai/gpt-6-sol |
2408.11016#3 | 2408.11016 | https://arxiv.org/abs/2408.11016 | math.CO | conjecture | critical_threshold | D | universality | existence_construction | null | What is the infimum of all ε>0 for which there exist γ>0 and n₀∈N such that, for every integer n≥n₀, every 2-edge-coloured 4-uniform hypergraph H on n vertices with δ₁(H)≥(365/512+ε) binom(n,3) contains a tight Hamilton cycle C with at least n/2+γn of its n edges in one colour? Here δ₁(H) is the minimum, over vertices ... | The answer is the infimum of the qualifying positive ε; it equals 0 exactly when the original assertion is true. | 0 | true | Is the following true? For every ε>0, do there exist γ>0 and n0 in N such that, for every integer n≥n0, every 2-edge-coloured 4-uniform hypergraph H on n vertices with minimum vertex degree δ1(H)≥(365/512+ε) binom(n,3) contains a tight Hamilton cycle C with at least n/2+γn of its n edges in one colour, where δ1(H) is t... | openai/gpt-6-sol |
2604.11937#6 | 2604.11937 | https://arxiv.org/abs/2604.11937 | math.CO | conjecture | critical_threshold | D | universality | bound_improvement | asymptotic_rate | For an integer k >= 3, let W_k be the graph obtained by joining one hub vertex to every vertex of a cycle of length k, so W_k has k+1 vertices. For a graph H, let R(H) be the least integer N such that every red/blue edge-coloring of K_N contains a monochromatic copy of H. Thus W_{2n} has 2n+1 vertices. For every intege... | Give one number: the infimum over eps > 0; a value of 0 means the original asymptotic claim holds. | 0 | true | For an integer k >= 3 let W_k denote the wheel on k+1 vertices, that is, the graph obtained by joining a single hub vertex to every vertex of a cycle of length k (note this indexing convention: W_k has k+1 vertices, not k). For a graph H let R(H) denote the diagonal 2-color Ramsey number of H, the least integer N such ... | openai/gpt-6-sol |
2605.25914#2 | 2605.25914 | https://arxiv.org/abs/2605.25914 | math.CO | conjecture | critical_threshold | D | universality | classification | null | For 3-uniform hypergraphs, let Pi^(3)_infty be the set of Turán densities pi(F)=lim_{n→∞} ex(n,F)/binom(n,3) of all possibly infinite families F of 3-graphs, where ex(n,F) is the maximum number of edges in an F-free 3-graph on n vertices. Call alpha in [0,1) a jump if there exists c>0 such that Pi^(3)_infty ∩ (alpha,al... | The value 4/9 means the original claim holds; a numerical answer is applicable only if such a threshold exists. | 4/9 | true | For 3-uniform hypergraphs, let Pi^(3)_infty be the set of Turan densities pi(F) of all possibly infinite families F of 3-graphs, where pi(F)=lim_{n->infinity} ex(n,F)/binom(n,3) and ex(n,F) is the maximum number of edges in an F-free 3-graph on n vertices. A number alpha in [0,1) is a jump if there exists c>0 such that... | openai/gpt-6-sol |
2503.03690#0 | 2503.03690 | https://arxiv.org/abs/2503.03690 | math.CO | conjecture | critical_threshold | D | universality | bound_improvement | null | Let p_* be the supremum of all real p for which there exists a constant c_p>0, depending only on p, such that |A+A|≥c_p N^p for every finite set A={a_1<⋯<a_N}⊂R whose consecutive differences are strictly increasing. What is p_*? | The answer is the single real number p_*. | 2 | true | Let A={a_1<...<a_N} be any finite subset of R whose consecutive differences a_{i+1}-a_i are strictly increasing for every 1<=i<N, and let A+A={a+a':a,a' in A}. Is it true that, for every epsilon>0, there exists a constant c_epsilon>0 depending only on epsilon such that |A+A|>=c_epsilon N^{2-epsilon}? | openai/gpt-6-sol |
2509.03490#3 | 2509.03490 | https://arxiv.org/abs/2509.03490 | math.CO | conjecture | critical_threshold | D | existence | bound_improvement | null | What is the supremum γ* of the values γ in (0, 1/10) for which the following holds: for every sufficiently large average degree d, every finite n-vertex graph G of average degree d whose adjacency-matrix eigenvalues are ordered λ₁≥⋯≥λₙ and satisfy |λₙ|=O(d^γ) contains a clique of order Ω(d/|λₙ|²)? Take the supremum of ... | Report one real number; γ*>0 means the original existence claim holds. | null | false | Does there exist a sufficiently small absolute constant gamma>0 such that, for every sufficiently large average degree d, every finite n-vertex graph G of average degree d whose adjacency-matrix eigenvalues are ordered lambda_1>=...>=lambda_n and satisfy |lambda_n|=O(d^gamma) contains a clique of order Omega(d/|lambda_... | openai/gpt-6-sol |
2503.05218#0 | 2503.05218 | https://arxiv.org/abs/2503.05218 | math.CO | conjecture | critical_threshold | D | universality | existence_construction | null | What is the infimum of the real numbers c>1 for which there exists N(c) such that, for all integers n≥N(c) and k≥2 with n>ck, every tripartite graph G with parts A, B, C satisfying |A|=|B|=|C|=n contains k vertex-disjoint triangles whenever α=|E(G[A,B])|/n², β=|E(G[A,C])|/n², and γ=|E(G[B,C])|/n² lie in [0,1] and satis... | Give the infimum as one value, taking it to be +∞ if no such c exists. | 1 | true | For every fixed constant C > 1, does there exist N(C) such that, for all integers n ≥ N(C) and k ≥ 2 with n > Ck, every tripartite graph G with parts A, B, C satisfying |A|=|B|=|C|=n contains k vertex-disjoint triangles whenever α=|E(G[A,B])|/n², β=|E(G[A,C])|/n², and γ=|E(G[B,C])|/n² lie in [0,1] and satisfy β(α-(k-1)... | openai/gpt-6-sol |
2512.12077#1 | 2512.12077 | https://arxiv.org/abs/2512.12077 | math.CO | conjecture | critical_threshold | D | universality | bound_improvement | null | Let s be an i.i.d. uniform word in {0,1}^omega. For each t>=0, let B_t be the set of words w for which the prefix s[1,t] has a partition of its positions into A_1,A_2 with s(A_1)=s(A_2)w. Define the boosted-greedy chain by X_0=T_0=0 and, whenever 0^{X_m} or 1^{X_m} lies in B_{T_m}, run one boosted-greedy cycle on the u... | The answer is a single number in [0,1/3]; the original question has answer Yes exactly when this number is 1/3. | 1/3 | true | Let s be an i.i.d. uniform word in {0,1}^omega, and for each t>=0 let B_t be the set of words w for which the prefix s[1,t] has a partition of its positions into A_1,A_2 with s(A_1)=s(A_2)w. Define the boosted-greedy chain by X_0=T_0=0 and, whenever 0^{X_m} or 1^{X_m} lies in B_{T_m}, run one boosted-greedy cycle on th... | openai/gpt-6-sol |
2605.29457#0 | 2605.29457 | https://arxiv.org/abs/2605.29457 | math.CO | conjecture | critical_threshold | D | existence | bound_improvement | null | What is the infimum of the exponents c>0 such that, for every finite non-abelian simple group G and every subset S of G for which the undirected Cayley graph Gamma(G;S) is connected—whose vertex set is G and in which distinct vertices g,h are adjacent exactly when g^{-1}h is in S or h^{-1}g is in S—the diameter of Gamm... | Give the critical exponent as one extended-real value, taking the infimum to be +∞ if no such c exists. | null | false | Does there exist a constant c>0 such that, for every finite non-abelian simple group G and every subset S of G for which the undirected Cayley graph Gamma(G;S) is connected—whose vertex set is G and in which distinct vertices g,h are adjacent exactly when g^{-1}h is in S or h^{-1}g is in S—the diameter of Gamma(G;S) is... | openai/gpt-6-sol |
2409.03128#0 | 2409.03128 | https://arxiv.org/abs/2409.03128 | math.CO | conjecture | critical_threshold | D | universality | bound_improvement | null | For every finite set A ⊂ R, let S_bi(A) be a largest subset S ⊂ A such that, for all a,b,a',b' ∈ S, each of a+b=a'+b' and ab=a'b' implies {a,b}={a',b'}. What is the supremum of the real numbers α such that, for every ε>0, all sufficiently large finite sets A ⊂ R satisfy |S_bi(A)| ≥ |A|^{α−ε}? | Give the critical exponent as one real number. | 1/2 | true | For every finite set A ⊂ R, let S_bi(A) be a largest subset S ⊂ A such that, for all a,b,a',b' ∈ S, each of the equalities a+b=a'+b' and ab=a'b' implies {a,b}={a',b'}. Theorem 1 proves |S_bi(A)| ≳ |A|^{1/3+7/78}; is it true, as |A|→∞, that every such A satisfies |S_bi(A)| ≥ |A|^{1/2+o(1)}? | openai/gpt-6-sol |
2409.19908#4 | 2409.19908 | https://arxiv.org/abs/2409.19908 | math.CO | conjecture | critical_threshold | D | universality | bound_improvement | asymptotic_rate | What is the infimum of the values ε>0 for which there exists N such that, for every integer n≥N and every 3-uniform hypergraph H on n vertices in which any two distinct edges intersect in at most one vertex, the maximum size α(H) of a vertex set containing no edge of H satisfies α(H)≥(1−ε)√(3n log n)? | The answer is the infimum of those ε>0; it is 0 exactly when the original assertion holds. | 0 | true | For every \varepsilon>0, does there exist N such that, for every integer n\ge N and every 3-uniform hypergraph H on n vertices in which any two distinct edges intersect in at most one vertex, the maximum size \alpha(H) of a vertex set containing no edge of H satisfies \alpha(H)\ge(1-\varepsilon)\sqrt{3n\log n}? | openai/gpt-6-sol |
2601.19879#0 | 2601.19879 | https://arxiv.org/abs/2601.19879 | math.CO | conjecture | critical_threshold | D | universality | bound_improvement | asymptotic_rate | For each integer d >= 3 and prime power q, let II_q^{(d)} be the bipartite graph between the points and affine lines of F_q^d, with adjacency given by incidence. An affine line is a set {x+tv : t in F_q}, where x is in F_q^d and v is a nonzero vector in F_q^d. Let IM(d,q) be the largest integer m for which there are po... | Give one real number; the answer is 0 exactly when the stated o(q^d) assertion holds for every fixed d >= 3. | 0 | true | Fix an integer d >= 3, and let q range over prime powers with d held fixed. For a prime power q let F_q^d be the d-dimensional vector space over the field with q elements, and call a set of the form l(x,v) = {x + tv : t in F_q}, with x in F_q^d and v in F_q^d \ {0}, an affine line. Let II_q^{(d)} be the bipartite incid... | openai/gpt-6-sol |
2604.16046#5 | 2604.16046 | https://arxiv.org/abs/2604.16046 | math.CO | conjecture | critical_threshold | D | universality | generalization | null | For each integer k >= 2, let c_k(G) be the minimum size of a multiset of paths of G that can be colored with k colors so that, for every two distinct edges e_1,e_2 of G, there are paths P_1,P_2 of different colors with e_1 in P_1 and e_2 not in P_1, and e_2 in P_2 and e_1 not in P_2. Let c_infty(G) be the minimum size ... | The answer is the infimum over positive epsilon satisfying the stated condition. | 0 | true | The class of all trees is known to be rainbow separable despite having unbounded pathwidth. Let Sigma be any infinite family of simple graphs for which there exists an integer t >= 0 such that every G in Sigma has treewidth at most t. For each integer k >= 2, let c_k(G) be the minimum size of a multiset of paths of G t... | openai/gpt-6-sol |
2509.02823#2 | 2509.02823 | https://arxiv.org/abs/2509.02823 | math.CO | conjecture | critical_threshold | D | universality | bound_improvement | null | Let K range over fields of characteristic 0, and let A range over finite subsets of K, with A+A={a+b:a,b in A} and A·A={ab:a,b in A}. What is the supremum of the real exponents α for which there exist c_α>0 and N_α such that every such K and A with |A|≥N_α satisfy max{|A+A|,|A·A|}≥c_α|A|^α? | The answer is one real number; it equals 2 exactly when the uniform |A|^{2-o(1)} strengthening holds. | 2 | true | Let K be any field with char(K)=0, and let A be any finite subset of K. Define A+A={a+b:a,b in A} and A·A={ab:a,b in A}. The currently established bound is max{|A+A|,|A·A|}\gtrsim |A|^{4/3}/(\log|A|)^{1/3}. Does the Erdős conjectural strengthening hold uniformly over such K: as |A|→∞, max{|A+A|,|A·A|}\gtrsim |A|^{2-o(1... | openai/gpt-6-sol |
2510.24691#8 | 2510.24691 | https://arxiv.org/abs/2510.24691 | math.CO | conjecture | critical_threshold | D | existence | bound_improvement | null | Let N(n,k,ell) be the maximum number of k-vertex subsets of an n-vertex graph that induce exactly ell edges, and let ind(k,ell)=lim_{n->infinity} N(n,k,ell)/binom(n,k). Theorem 1.3 gives the sparse-regime upper-bound scale O(ell^{-1/4}), while for each ell=binom(m,2)<=k there are constructions with ind(k,ell) ≳ ell^{-1... | Give the supremum as one number, taking it to be 0 if no positive delta qualifies; the original existence claim holds exactly when it is positive. | null | false | Let N(n,k,ell) be the maximum number of k-vertex subsets of an n-vertex graph that induce exactly ell edges, and define the edge-inducibility ind(k,ell)=lim_{n->infinity} N(n,k,ell)/binom(n,k). Theorem 1.3 gives the sparse-regime upper-bound scale O(ell^{-1/4}), and for each ell=binom(m,2)<=k there are constructions wi... | openai/gpt-6-sol |
2412.14891#8 | 2412.14891 | https://arxiv.org/abs/2412.14891 | math.CO | conjecture | critical_threshold | D | universality | bound_improvement | existence_construction | What is the infimum of all ε > 0 for which there exists n0 such that, for every integer n ≥ n0, every n-vertex 3-uniform hypergraph G with minimum codegree δ2(G) := min{|{v ∈ V(G) : S ∪ {v} ∈ E(G)}| : S ∈ (V(G) choose 2)} satisfying δ2(G) ≥ (3/4 + ε)n contains a square of a tight Hamilton cycle—that is, a cyclic orderi... | The answer is this infimum; it equals 0 exactly when the original statement holds. | 0 | true | For every ε > 0, does there exist n0 such that, for every integer n ≥ n0, every n-vertex 3-uniform hypergraph G with minimum codegree δ2(G) := min{|{v ∈ V(G) : S ∪ {v} ∈ E(G)}| : S ∈ (V(G) choose 2)} satisfying δ2(G) ≥ (3/4 + ε)n contain a square of a tight Hamilton cycle; that is, a cyclic ordering of all n vertices s... | openai/gpt-6-sol |
2509.01568#1 | 2509.01568 | https://arxiv.org/abs/2509.01568 | math.CO | conjecture | critical_threshold | D | universality | bound_improvement | asymptotic_rate | Let A={a_1<...<a_n} be any finite subset of R whose consecutive differences a_{i+1}-a_i are strictly increasing, and let E(A)=|{(a,b,c,d) in A^4:a+b=c+d}|. What is the critical exponent p_* = inf{p in R: for every ε>0, there is an N such that E(A)≤n^{p+ε} for every n≥N and every such A of size n}? | The answer is the single real number p_*. | 2 | true | Let A={a_1<...<a_n} be any finite subset of R whose consecutive differences a_{i+1}-a_i are strictly increasing for every 1<=i<n, and define its additive energy by E(A)=|{(a,b,c,d) in A^4:a+b=c+d}|. Is it true, as n=|A| tends to infinity, that E(A)<=|A|^{2+o(1)}? Equivalently in asymptotic-exponent terms, can every suc... | openai/gpt-6-sol |
2508.04480#0 | 2508.04480 | https://arxiv.org/abs/2508.04480 | math.CO | conjecture | critical_threshold | D | universality | bound_improvement | null | Let G be a totally ordered abelian group, and let A={a_1<...<a_n} be a finite subset of G with n=|A| and a_{i+2}-a_{i+1}>a_{i+1}-a_i for every 1<=i<=n-2. Write A+A={a+a': a,a' in A}. Using the paper's Vinogradov convention allowing suppressed logarithmic factors, what is the supremum of the exponents 2-epsilon, over ep... | Answer with the supremum exponent; the value 2 means the original assertion holds for every epsilon>0. | 2 | true | Let G be a totally ordered abelian group and let A={a_1<...<a_n} be a finite subset of G, where n=|A| and a_{i+2}-a_{i+1}>a_{i+1}-a_i for every 1<=i<=n-2. Writing A+A={a+a': a,a' in A}, prove that for every epsilon>0 there is a constant c_epsilon>0 such that |A+A|>=c_epsilon|A|^{2-epsilon} for every such convex set A, ... | openai/gpt-6-sol |
2603.28202#0 | 2603.28202 | https://arxiv.org/abs/2603.28202 | math.CO | conjecture | critical_threshold | D | extension | bound_improvement | null | Theorem 1.1 establishes the following conclusion with minimum codegree at least (7/9+alpha)n. Let alpha_* be the infimum of all real alpha>0 for which there exists n0 in N such that, for every integer n>=n0, every 3-uniform hypergraph H on n vertices with delta_2(H)>=(3/4+alpha)n has a cyclic ordering of V(H) in which ... | Give one real number; alpha_*=0 means the proposed conclusion holds for every alpha>0. | 0 | true | Theorem 1.1 proves that for every alpha>0 and all sufficiently large n, every 3-uniform hypergraph H on n vertices with minimum codegree at least (7/9+alpha)n contains the square of a tight Hamilton cycle. Can this be improved to the following asymptotically sharp statement: for every alpha>0, does there exist n0 in N ... | openai/gpt-6-sol |
2510.22234#2 | 2510.22234 | https://arxiv.org/abs/2510.22234 | math.CO | conjecture | critical_threshold | D | universality | bound_improvement | existence_construction | What is the critical value R=sup_G Φ₂^∞(G), where G ranges over all bridgeless graphs and Φ₂^∞(G) is the minimum r≥2 for which there are an orientation of E(G) and a map φ:E(G)→ℝ² such that, at every vertex, the sum of the vectors on incoming edges equals the sum on outgoing edges, and 1≤‖φ(e)‖∞≤r−1 for every edge e, w... | The answer is the supremum R; the original question has answer Yes exactly when R≤5/2. | null | false | For every bridgeless graph G, is there an orientation of E(G) and a map \varphi:E(G)\to\mathbb{R}^2 such that, at each vertex, the sum of the vectors on incoming edges equals the sum on outgoing edges, and every edge e satisfies 1\leq\|\varphi(e)\|_\infty\leq3/2, where \|(x_1,x_2)\|_\infty=\max\{|x_1|,|x_2|\}? Equivale... | openai/gpt-6-sol |
1705.00990#2 | 1705.00990 | https://arxiv.org/abs/1705.00990 | math.CO | conjecture | critical_threshold | D | universality | bound_improvement | existence_construction | What is the infimum of all c ∈ [0, 2/3] such that, for every real γ > c, there exists n₀ such that every integer n ≥ n₀ divisible by 3 and every 3-uniform hypergraph H on n vertices with no copy of K₄³ and with δ₂(H) ≥ (1/3 + γ)n has a perfect matching? Here K₄³ is the complete 3-uniform hypergraph on four vertices, δ₂... | The answer is the critical value of γ above which the stated eventual perfect-matching property holds. | 0 | true | For every real γ > 0, does there exist n_0 such that, for every integer n ≥ n_0 with 3 dividing n, every 3-uniform hypergraph H on n vertices satisfying the following conditions has a perfect matching: (i) H contains no copy of K_4^3, the complete 3-uniform hypergraph on four vertices; and (ii) for every pair S of dist... | openai/gpt-6-sol |
2411.18782#6 | 2411.18782 | https://arxiv.org/abs/2411.18782 | math.CO | conjecture | critical_threshold | D | existence | bound_improvement | asymptotic_rate | For each positive integer n, let T'(n)={τ(G): G is a simple, not necessarily planar, graph with n vertices}, where τ(G) is the number of spanning trees of G. What is the critical value a_*=liminf_{n→∞} log|T'(n)|/(n log n)? | Give the exact value of a_*; the original assertion holds exactly when a_*>0. | null | false | For each positive integer n, let T'(n) = {tau(G) : G is a simple graph with n vertices}, where tau(G) denotes the number of spanning trees of G; thus |T'(n)| is the number of distinct spanning-tree numbers attained by all simple, not necessarily planar, n-vertex graphs. The planar main theorem already implies |T'(n)| >... | openai/gpt-6-sol |
End of preview. Expand in Data Studio
ResearchMath-2-6Sol-HillClimb
1,694 open research-mathematics questions whose answer is a number you can make
incremental progress toward. A subset of
amphora/ResearchMath-2-6Sol-Rewrite
(numerical config), keeping only the two question types with a continuous target.
| config | rows | the answer is |
|---|---|---|
sharp_constant |
859 | the best constant in an inequality or bound |
critical_threshold |
835 | the critical value of a parameter where behaviour changes |
default |
1,694 | both |
Why these two types
Most open questions are binary — is there a counterexample? — and admit no partial credit. The questions here instead ask for a value, so every improvement on the best known bound is real, measurable progress:
- Sharp constant. "What is the supremum of the constants $c>0$ such that every graph with $\alpha^(G)<m$ contains a $K_{\lfloor cn/\sqrt m\rfloor}$-minor?"* — each proof that a larger $c$ works (or a construction showing a smaller $c$ fails) narrows the answer.
- Critical threshold. "What is $\inf{c>\sqrt{\log 2} : S_c \text{ has asymptotic density } 0}$?" — each result pushing the proven threshold down moves toward the conjectured value.
Conjectured endpoints
has_conjectured_endpoint is true for 667 rows (195 sharp-constant, 472 critical-threshold),
where the source paper conjectures the answer and distinguished_value records it. For these,
progress can be measured as the remaining gap to a known target. The rest have an open target
with no conjectured value.
⚠️ Progress is not automatically checkable
A continuous target does not by itself give a cheap reward signal. What matters is how progress is verified, and that depends on direction:
- By construction — exhibiting an explicit graph, sequence or example — progress can be checked by computer.
- By proof — showing a statement holds for every object — progress needs a proof checker (e.g. Lean) or a human.
For a sharp constant, the valuable direction (showing a larger constant works) is usually a proof, while the opposite direction (a counterexample showing a constant fails) is usually a construction. Critical-threshold questions about asymptotic behaviour typically need proofs in both directions. Construction-verifiable questions are the cheap-reward subset and are likely concentrated in combinatorics; this dataset does not yet label which is which.
⚠️ There are no answer keys
These are open problems taken from future-work and open-problem statements. Their answers are,
by construction, not known. distinguished_value is a conjecture from the source paper, not a
verified answer.
Composition
math_intent (from the parent dataset):
| math_intent | rows |
|---|---|
bound_improvement |
1,192 |
existence_construction |
157 |
asymptotic_rate |
108 |
generalization |
86 |
hypothesis_weakening |
67 |
structural_characterization |
48 |
algorithmic |
16 |
classification |
10 |
other |
8 |
uniqueness |
2 |
Top arXiv categories:
| category | rows |
|---|---|
math.CO |
687 |
math.NT |
299 |
math.CA |
87 |
math.AP |
74 |
math.PR |
72 |
math.OC |
38 |
math.MG |
37 |
math-ph |
37 |
Fields
| field | description |
|---|---|
uid |
<arxiv_id>#<index>, joins to the parent datasets |
arxiv_id, paper_url |
source paper |
primary_category |
arXiv category |
signal_type |
source statement type (conjecture, open_problem, …) |
target_type |
sharp_constant or critical_threshold |
route |
C (sharp constant) or D (critical threshold) — the parent dataset's conversion route |
question_category |
logical shape of the source question |
math_intent, secondary_intent |
what the source question is about |
question |
the reformatted question |
answer_convention |
what the number means |
distinguished_value |
the source's conjectured answer, or null |
has_conjectured_endpoint |
whether distinguished_value is set |
source_question |
the unmodified original |
engine |
openai/gpt-6-sol |
Caveats
- Reformatting can strengthen the problem. Asking for the best constant is strictly harder than asking whether some constant works, which is often what the source asked.
- Some source statements are mis-transcribed. A literature search over a sample of the parent dataset found ~8% of questions where the source paper dropped a hypothesis or had a typo, making the literal question degenerate. That search covered few questions in this subset, so such items may remain here.
- A small number of
arxiv_idvalues are wrong, inherited from the source corpus.
Licensing
Questions and derived fields: CC-BY-4.0. No verbatim paper excerpts are included.
Citation
@article{son2026researchmath,
title={ResearchMath-14K: Scaling Research-Level Mathematics via Agents},
author={Son, Guijin and Yi, Seungyeop and Gwak, Minju and Ko, Hyunwoo and Jang, Wongi and Yu, Youngjae},
journal={arXiv preprint arXiv:2605.28003},
year={2026}
}
Collaborations
I'm interested in creating larger datasets to train open models for research-level math. If you are interested let me know. (guijin.son@snu.ac.kr)
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