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In the special case \( {U}_{1}\left( {t, x}\right) = {U}_{2}\left( x\right) = \log x,0 \leq t \leq T, x > 0 \), we have \( {\widetilde{U}}_{1}\left( {t, y}\right) = {\widetilde{U}}_{2}\left( y\right) = \) \( - \left( {1 + \log y}\right), y > 0 \) (see Example 4.2), and | \[ {\widetilde{V}}_{\nu }\left( y\right) = - \left( {T + 1}\right) \left( {1 + \log y}\right) - {\int }_{0}^{T}E\left\lbrack {\log {H}_{\nu }\left( t\right) }\right\rbrack {dt} - E\left\lbrack {\log {H}_{\nu }\left( T\right) }\right\rbrack . \] For \( \nu \left( \cdot \right) \in \mathcal{D} \), we have \[ - E\left\lbr... | Yes |
Lemma 1.1 (Markovian property of LERW). Consider \( {y}_{0},\ldots ,{y}_{j} \in \mathcal{S} \) so that with positive probability for \( \mathcal{L}\left( {x,{y}_{0};A}\right) \) , \[ \left\{ {{L}_{\sigma } = {y}_{0},{L}_{\sigma - 1} = {y}_{1},\ldots ,{L}_{\sigma - j} = {y}_{j}}\right\} . \] The conditional law of \( L\... | Proof. For each \( A \) and \( x \in \mathcal{A} \), we denote by \( G\left( {x, A}\right) \) the expected number of visits by the Markov chain \( X \) before \( {\tau }_{A} \) if \( {X}_{0} = x \) . Then, it is a simple exercise to check that for all \( n \geq 1,\mathbf{w} = \left( {{w}_{0},\ldots ,{w}_{n}}\right) \) ... | Yes |
Proposition 2.1 The following two conditions are equivalent:\n\n1. \( \\left( {{K}_{t}, t \\geq 0}\\right) \) is a Loewner chain associated to a continuous driving function \( \\left( {{w}_{t}, t \\geq 0}\\right) \) .\n\n2. For all \( t \\geq 0, a\\left( {K}_{t}\\right) = t \), and for all \( T > 0 \), and \( \\varepsi... | Sketch of the proof. Let us now prove that 2. implies 1. (the fact that 1. implies 2. is very easy): 2. implies that for all \( t \\geq 0 \), the diameter of the sets \( {g}_{t}\\left( {{K}_{t + \\delta } \\smallsetminus {K}_{t}}\\right) \) decrease towards 0 when \( \\delta \\rightarrow 0 \) . Hence, one can simply de... | No |
Proposition 3.4 The law of \( X \) is the uniform distribution on \( {AU} \) . | This is a direct consequence of the explicit computation of \( \widetilde{F} \) and of the explicit Schwarz-Christoffel mapping from the upper half-plane onto \( \mathcal{T} \) : For each \( C \in {AU} \), one can compute the probability that \( X \in \left\lbrack {AC}\right\rbrack \) via the function \( \widetilde{F} ... | No |
Proposition 3.5 For \( \kappa < 4 \), almost surely, \( \mathop{\lim }\limits_{{t \rightarrow \infty }}{\gamma }_{t} = \infty \) . | Proof. Let \( \delta \in \left( {0,1/4}\right), x > 1 \), and suppose that\n\n\[ \n{t}_{\delta } \mathrel{\text{:=}} \inf \left\{ {t > 0 : d\left( {{\gamma }_{t},\left\lbrack {1, x}\right\rbrack }\right) \leq \delta }\right\} \n\]\n\nis finite. Let \( {z}_{\delta } = {\gamma }_{{t}_{\delta }} \) . Clearly, \( {g}_{{t}_... | No |
Proposition 4.2 Consider chordal \( {SL}{E}_{8/3} \) in \( \mathbb{H} \) . Then, for any Hull \( A \) ,\n\n\[ \n\mathbf{P}\left\lbrack {\forall t \geq 0,{K}_{t} \cap A = \varnothing }\right\rbrack = {\Phi }_{A}^{\prime }{\left( 0\right) }^{5/8}.\n\] | Proof. The quantity \( {M}_{t} \mathrel{\text{:=}} {h}_{t}^{\prime }{\left( {W}_{t}\right) }^{5/8} \) is a local martingale. Recall that \( {h}_{t} \) is a normalized map from a subset of \( \mathbb{H} \) onto \( \mathbb{H} \) . Hence, for all \( t < T,{M}_{t} \leq 1 \) and \( M \) is a bounded martingale. We have to u... | Yes |
Corollary 4.3. Suppose that \( {A}_{0} \) is a Hull, then the conditional law of \( {K}_{\infty } \) given \( {K}_{\infty } \cap {A}_{0} = \varnothing \) is identical to the law of \( {\Psi }_{{A}_{0}}^{-1}\left( {K}_{\infty }\right) \) . | Proof. Note that \( {K}_{\infty } \) is a closed set because of the transience of \( \left( {{K}_{t}, t \geq }\right. \) \( 0) \) . The law of such a random set is characterized by the value of \( \mathbf{P}\left\lbrack {{K}_{\infty } \cap }\right. \) \( A = \varnothing \rbrack \) for all Hulls \( A \) (this set of eve... | Yes |
Theorem 4.3. Chordal \( {SL}{E}_{8/3} \) is the unique measure on continuous simple curves \( \gamma \) from 0 to \( \infty \) in \( \mathbb{H} \) such that for all Hull, the law of \( \gamma \) conditioned to avoid \( A \) is identical to the law of \( {\Psi }^{-1}\left( \gamma \right) \) . | The proof of this Theorem uses the complete description of all measures on simply connected closed sets (not necessarily curves) joining 0 to \( \infty \) in \( \mathbb{H} \) that satisfy this condition. These measures (called restriction measures in 95) are constructed using \( {SL}{E}_{\kappa } \) (in fact, by adding... | Yes |
Lemma 5.2. Suppose that \( \Phi \) is the conformal transformation from \( \mathbb{H} \) onto an equilateral triangle \( {OAC} \) such that \( \Phi \left( 0\right) = O,\Phi \left( {-1}\right) = A \) and \( \Phi \left( 1\right) = C \) . Then, the law of \( \Phi \left( {Z}_{{\sigma }_{\mathbb{H}}}\right) \) is uniform on... | Proof. One elementary convincing proof uses discrete approximations. Here is a brief outline of this proof: Define \( \omega = \exp \left( {{i\pi }/3}\right) \) . Consider a triangular grid in the wedge \( \mathcal{W} \) i.e. \( \left\{ {m + {m}^{\prime }\omega : m,{m}^{\prime } \geq 0}\right\} \) . Let \( \left( {{S}_... | Yes |
Theorem 5.1. Define the following two sets:\n\n- Consider chordal \( {SL}{E}_{6}\left( {{K}_{t}, t \geq 0}\right) \) in \( \mathbb{H} \) (or in \( V \) ) up to its first hitting time \( T \) of \( \mathbb{R} \smallsetminus \left( {-1,1}\right) \) . Let \( e \) denote the point at which the SLE hits \( \mathbb{R} \small... | Proof. Note that Lemma 5.2 Lemma 5.1 Theorem 4.1 and Proposition 3.4 show that \( E \) and \( F \) both have the following properties:\n\n- They are random compact sets that intersect \( \mathbb{R} \smallsetminus \left( {-1,1}\right) \) at just one point \( x \) and the law of \( \Phi \left( x\right) \) is uniform on \... | Yes |
Theorem 5.2. Suppose that \( {\mathcal{H}}_{8} \) denotes the filling of the union of 8 independent chordal \( {SL}{E}_{8/3} \) ’s. Suppose that \( {\mathcal{H}}_{5} \) denotes the filling of the union of 5 independent Brownian excursions. Then, \( {\mathcal{H}}_{5} \) and \( {\mathcal{H}}_{8} \) have the same law. | Proof. This is simply due to the fact that for all Hull \( A \)\n\n\[ \mathbf{P}\left\lbrack {{\mathcal{H}}_{5} \cap A = \varnothing }\right\rbrack = \mathbf{P}\left\lbrack {{\mathcal{H}}_{8} \cap A = \varnothing }\right\rbrack = {\Phi }_{A}^{\prime }{\left( 0\right) }^{5} \]\n\nand that this characterizes these laws. | No |
Proposition 6.2 Let \( \left( {{K}_{t}, t \geq 0}\right) ,\left( {{\widetilde{K}}_{u}, u \geq 0}\right), T \) and \( \widetilde{T} \) be defined just as in Theorem 6.1, except that they are SLE with general \( \kappa > 0 \) . There exist two nondecreasing families of stopping times \( \left( {{T}_{n}, n \geq 1}\right) ... | Proof. Let us first briefly recall how \( {\widetilde{K}}_{u} \) is defined. For convenience, we will restrict ourselves to \( x = \pi \) (the proof in the general case is almost identical). Define the conformal map\n\n\[ \psi \left( z\right) = i\frac{1 - z}{1 + z} \]\n\nfrom \( \mathbb{U} \) onto \( \mathbb{H} \) that... | Yes |
Theorem 6.3. Suppose that \( r < 1 \) . Define the two following random hulls:\n\n- Suppose that \( \left( {{K}_{t}, t \geq 0}\right) \) is radial \( {SL}{E}_{6} \) as before. Let \( {\tau }_{r} \) denote the first time at which radial \( {K}_{t} \) intersects the circle \( \{ \left| z\right| = r\} \) . Define the even... | In particular,\n\n\[ \mathbf{P}\left\lbrack {\mathcal{H}\left( {x,{\tau }_{r}}\right) }\right\rbrack = \mathbf{P}\left\lbrack {\widetilde{\mathcal{H}}\left( {x,{\widetilde{\sigma }}_{r}}\right) )}\right\rbrack \]\n\nThis shows that one can compute non-disconnection probabilities for reflecting Brownian motions using ra... | Yes |
Lemma 8.1. There exists \( \delta > 0 \) and \( \varepsilon > 0 \) such that for all integer \( n \), then for at least \( {99}\% \) of the integers \( j \in \{ 1,\ldots, n\} \), one has\n\n\[ \n{p}_{{r}_{j}}^{ * }\left( \delta \right) > \varepsilon {p}_{{r}_{j}}^{ * }\n\]\n\nwhere \( {r}_{j} = {2}^{-j} \) . | Proof. We only sketch the main ideas of the proof. First, notice that \( j \mapsto \) \( {p}_{{r}_{j}}^{ * } \) is decreasing in \( j \) so that the a priori lower bound for \( {p}_{{r}_{j}}^{ * } \) implies that there exists \( \varepsilon \) such that for all \( n \), then for at least \( {99}\% \) of the values of \... | Yes |
Lemma 8.2. For all fixed \( \delta \), for some constant \( c = c\left( \delta \right) \) ,\n\n\[ \mathbf{P}\left\lbrack {{P}_{r}^{1} \subset \mathcal{B}\left( {1,\delta /2}\right) }\right\rbrack \geq \frac{c}{\log \left( {1/r}\right) }.\] | Proof. With positive probability, \( Z \) hits the circle of radius \( 1 - \delta /4 \) around 0 before \( \partial \mathcal{B}\left( {1,\delta /2}\right) \) . Then, if this is the case, with probability \( \log (1/(1 - \) \( \delta /4))/\log \left( {1/r}\right) \) it hits the circle of radius \( r \) before going back... | No |
Theorem 8.1. One has \( \eta = 1/4 \) . Furthermore, there exist two constants \( {c}_{1} \) and \( {c}_{2} \) such that for all \( R > 1 \) , \[ {c}_{1}{R}^{-1/4} \leq {p}_{R} \leq {c}_{2}{R}^{-1/4}. \] | Proof. By inversion, this is equivalent to corresponding result for small \( r \) i.e., that for all \( r < 1 \) , \[ {c}_{1}{r}^{1/4} \leq {p}_{r} \leq {c}_{2}{r}^{1/4} \] (8.1) In order to compare \( {p}_{r} \) to \( {\widetilde{p}}_{r} \) (this is the non-disconnection probability for reflected Brownian motion that ... | No |
Proposition 9.3 (Wilson’s algorithm) The law of \( {A}_{m} \) is the uniform spanning tree measure. | Proof. One can derive this result using the explicit formulas that we derived in the introductory chapter for loop-erased random walks: Indeed, it follows readily from the definition and the symmetry of the function \( F \) that was defined there, and the fact that (since we are considering simple random walks), the tr... | Yes |
Theorem 10.1. Cardy's prediction is true in the case of critical site percolation on the triangular lattice. | In fact, Smirnov's proof is a direct proof of Cardy's formula that does not rely at all on SLE. Then, with Smirnov's result, one can show that indeed the scaling limit of the percolation exploration process is \( {SL}{E}_{6} \) . Sketch of the proof. Suppose first for convenience that \( {AOC} \) is an equilateral tria... | Yes |
Theorem 10.3. If one performs site percolation on the triangular lattice with probability \( p \), then when \( p \rightarrow 1/2 + \), the probability that the origin belongs to the infinite cluster behaves like \( {\left( p - 1/2\right) }^{5/{36} + o\left( 1\right) } \) . When \( p \rightarrow 1/2 - \), the correlati... | See [68 128 for more results as well as for the proofs... | No |
Proposition 1.1.1 If \( \left( \begin{array}{l} n \\ k \end{array}\right) \cdot {2}^{1 - \left( \begin{array}{l} k \\ 2 \end{array}\right) } < 1 \) then \( R\left( {k, k}\right) > n \) . Thus \( R\left( {k, k}\right) > \left\lfloor {2}^{k/2}\right\rfloor \) for all \( k \geq 3 \) . | Proof. Consider a random two-coloring of the edges of \( {K}_{n} \) obtained by coloring each edge independently either red or blue, where each color is equally likely. For any fixed set \( R \) of \( k \) vertices, let \( {A}_{R} \) be the event that the induced subgraph of \( {K}_{n} \) on \( R \) is monochromatic (i... | Yes |
Theorem 1.2.1 If \( \left( \begin{array}{l} n \\ k \end{array}\right) {\left( 1 - {2}^{-k}\right) }^{n - k} < 1 \) then there is a tournament on \( n \) vertices that has the property \( {S}_{k} \) . | Proof. Consider a random tournament on the set \( V = \{ 1,\ldots, n\} \) . For every fixed subset \( K \) of size \( k \) of \( V \), let \( {A}_{K} \) be the event that there is no vertex which beats all the members of \( K \) . Clearly \( \Pr \left( {A}_{K}\right) = {\left( 1 - {2}^{-k}\right) }^{n - k} \) . This is... | Yes |
Theorem 1.2.2 Let \( G = \left( {V, E}\right) \) be a graph on \( n \) vertices, with minimum degree \( \delta > 1 \) . Then \( G \) has a dominating set of at most \( n\frac{1 + \ln \left( {\delta + 1}\right) }{\delta + 1} \) vertices. | Proof. Let \( p \in \left\lbrack {0,1}\right\rbrack \) be, for the moment, arbitrary. Let us pick, randomly and independently, each vertex of \( V \) with probability \( p \) . Let \( X \) be the (random) set of all vertices picked and let \( Y = {Y}_{X} \) be the random set of all vertices in \( V - X \) that do not h... | Yes |
Lemma 1.2.3 Let \( G = \left( {V, E}\right) \) be a graph with minimum degree \( \delta \) and let \( V = {V}_{1} \cup {V}_{2} \) be a cut of size smaller than \( \delta \) in \( G \) . Then every dominating set \( U \) of \( G \) has vertices in \( {V}_{1} \) and in \( {V}_{2} \) . | Proof. Suppose this is false and \( U \subseteq {V}_{1} \) . Choose, arbitrarily, a vertex \( v \in {V}_{2} \) and let \( {v}_{1},{v}_{2},\ldots ,{v}_{\delta } \) be \( \delta \) of its neighbors. For each \( i,1 \leq i \leq \delta \), define an edge \( {e}_{i} \) of the given cut as follows; if \( {v}_{i} \in {V}_{1} ... | Yes |
Proposition 1.3.1 [Erdős (1963a) ] Every n-uniform hypergraph with less than \( {2}^{\;n - 1} \) edges has property \( B \) . Therefore \( m\left( n\right) \geq {2}^{n - 1} \) . | Proof. Let \( H = \left( {V, E}\right) \) be an \( n \) -uniform hypergraph with less than \( {2}^{n - 1} \) edges. Color \( V \) randomly by 2 colors. For each edge \( e \in E \), let \( {A}_{e} \) be the event that \( e \) is monochromatic. Clearly \( \Pr \left( {A}_{e}\right) = {2}^{1 - n} \) . Therefore\n\n\[ \Pr \... | Yes |
Theorem 1.3.3 If \( \mathcal{F} = {\left\{ \left( {A}_{i},{B}_{i}\right) \right\} }_{i = 1}^{h} \) is a \( \left( {k,\ell }\right) \) -system then \( h \leq \left( \begin{matrix} k + \ell \\ k \end{matrix}\right) \) . | Proof. Put \( X = \mathop{\bigcup }\limits_{{i = 1}}^{h}\left( {{A}_{i} \cup {B}_{i}}\right) \) and consider a random order \( \pi \) of \( X \) . For each \( i \) , \( 1 \leq i \leq k \), let \( {X}_{i} \) be the event that all the elements of \( {A}_{i} \) precede all those of \( {B}_{i} \) in this order. Clearly \( ... | Yes |
Every set \( B = \left\{ {{b}_{1},\ldots ,{b}_{n}}\right\} \) of \( n \) nonzero integers contains a sum-free subset \( A \) of size \( \left| A\right| > \frac{1}{3}n \) . | Proof. Let \( p = {3k} + 2 \) be a prime, which satisfies \( p > 2\mathop{\max }\limits_{{1 \leq i \leq n}}\left| {b}_{i}\right| \) and put \( C = \) \( \{ k + 1, k + 2,\ldots ,{2k} + 1\} \) . Observe that \( C \) is a sum-free subset of the cyclic group \( {Z}_{p} \) and that \( \frac{\left| C\right| }{p - 1} = \frac{... | Yes |
Theorem 1.5.1 Let \( \mathcal{F} \) be a family of \( m = {2}^{\left( {\frac{1}{2} + \delta }\right) n} \) subsets of \( X = \{ 1,2,\ldots, n\} \) , where \( \delta > 0 \) . Then\n\n\[ d\left( \mathcal{F}\right) < {m}^{2 - \frac{{\delta }^{2}}{2}}. \] | Proof. Suppose (1.1) is false and pick independently \( t \) members \( {A}_{1},{A}_{2},\ldots ,{A}_{t} \) of \( \mathcal{F} \) with repetitions at random, where \( t \) is a large positive integer, to be chosen later.\n\nWe will show that with positive probability \( \left| {{A}_{1} \cup {A}_{2} \cup \ldots \cup {A}_{... | Yes |
Lemma 1 For \( 0 \leq s \leq n - 1 \) set \( {A}_{s} = \{ s, s + 1,\ldots, s + k - 1\} \) where addition is modulo \( n \) . Then \( \mathcal{F} \) can contain at most \( k \) of the sets \( {A}_{s} \) . | Proof. Fix some \( {A}_{s} \in \mathcal{F} \) . All other sets \( {A}_{t} \) that intersect \( {A}_{s} \) can be partitioned into \( k - 1 \) pairs \( \left\{ {{A}_{s - i},{A}_{s + k - i}}\right\} ,\left( {1 \leq i \leq k - 1}\right) \), and the members of each such pair are disjoint. The result follows, since \( \math... | Yes |
Theorem 2.1.1 There is a tournament \( T \) with \( n \) players and at least \( n!{2}^{-\left( {n - 1}\right) } \) Hamiltonian Paths. | Proof. In the random tournament let \( X \) be the number of Hamiltonian paths. For each permutation \( \sigma \) let \( {X}_{\sigma } \) be the indicator random variable for \( \sigma \) giving a Hamiltonian path - i.e., satisfying \( \left( {\sigma \left( i\right) ,\sigma \left( {i + 1}\right) }\right) \in T \) for \... | Yes |
Theorem 2.2.1 Let \( G = \\left( {V, E}\\right) \) be a graph with \( n \) vertices and \( e \) edges. Then \( G \) contains a bipartite subgraph with at least \( e/2 \) edges. | Proof. Let \( T \\subseteq V \) be a random subset given by \( \\Pr \\left\\lbrack {x \\in T}\\right\\rbrack = 1/2 \), these choices mutually independent. Set \( B = V - T \) . Call an edge \( \\{ x, y\\} \) crossing if exactly one of \( x, y \) are in \( T \) . Let \( X \) be the number of crossing edges. We decompose... | Yes |
Theorem 2.2.2 If \( G \) has \( {2n} \) vertices and e edges then it contains a bipartite subgraph with at least \( \frac{en}{{2n} - 1} \) edges. If \( G \) has \( {2n} + 1 \) vertices and e edges then it contains a bipartite subgraph with at least \( \frac{e\left( {n + 1}\right) }{{2n} + 1} \) edges. | Proof. When \( G \) has \( {2n} \) vertices let \( T \) be chosen uniformly from among all \( n \) -element subsets of \( V \) . Any edge \( \{ x, y\} \) now has probability \( \frac{n}{{2n} - 1} \) of being crossing and the proof concludes as before. When \( G \) has \( {2n} + 1 \) vertices choose \( T \) uniformly fr... | No |
Lemma 2.2.4 Let \( {P}_{k} \) denote the set of all homogeneous polynomials \( f\left( {{p}_{1},\ldots ,{p}_{k}}\right) \) of degree \( k \) with all coefficients having absolute value at most one and \( {p}_{1}{p}_{2}\cdots {p}_{k} \) having coefficient one. Then for all \( f \in {P}_{k} \) there exist \( {p}_{1},\ldo... | Proof. Set\n\n\[ M\left( f\right) = \mathop{\max }\limits_{{{p}_{1},\ldots ,{p}_{k} \in \left\lbrack {0,1}\right\rbrack }}\left| {f\left( {{p}_{1},\ldots ,{p}_{k}}\right) }\right| \]\n\nFor \( f \in {P}_{k}, M\left( f\right) > 0 \) as \( f \) is not the zero polynomial. As \( {P}_{k} \) is compact and \( M : {P}_{k} \r... | Yes |
Theorem 2.3.1 There is a two-coloring of \( {K}_{n} \) with at most\n\n\[ \left( \begin{array}{l} n \\ a \end{array}\right) {2}^{1 - \left( \begin{array}{l} a \\ 2 \end{array}\right) } \]\n\nmonochromatic \( {K}_{a} \) . | Proof.[outline] Take a random coloring . Let \( X \) be the number of monochromatic \( {K}_{a} \) and find \( E\left\lbrack X\right\rbrack \) . For some coloring the value of \( X \) is at most this expectation. - | No |
Theorem 2.3.2 There is a two-coloring of \( {K}_{m, n} \) with at most\n\n\[ \left( \begin{matrix} m \\ a \end{matrix}\right) \left( \begin{array}{l} n \\ b \end{array}\right) {2}^{1 - {ab}} \]\n\nmonochromatic \( {K}_{a, b} \) . | Proof.[outline] Take a random coloring . Let \( X \) be the number of monochromatic \( {K}_{a, b} \) and find \( E\left\lbrack X\right\rbrack \) . For some coloring the value of \( X \) is at most this expectation. ∎ | No |
Theorem 2.4.1 Let \( {v}_{1},\ldots ,{v}_{n} \in {R}^{n} \), all \( \left| {v}_{i}\right| = 1 \) . Then there exist \( {\epsilon }_{1},\ldots ,{\epsilon }_{n} = \pm 1 \) so that\n\n\[ \left| {{\epsilon }_{1}{v}_{1} + \ldots + {\epsilon }_{n}{v}_{n}}\right| \leq \sqrt{n} \]\n\nand also there exist \( {\epsilon }_{1},\ld... | Proof. Let \( {\epsilon }_{1},\ldots ,{\epsilon }_{n} \) be selected uniformly and independently from \( \{ - 1, + 1\} \) . Set\n\n\[ X = {\left| {\epsilon }_{1}{v}_{1} + \ldots + {\epsilon }_{n}{v}_{n}\right| }^{2} \]\n\nThen\n\n\[ X = \mathop{\sum }\limits_{{i = 1}}^{n}\mathop{\sum }\limits_{{j = 1}}^{n}{\epsilon }_{... | Yes |
Theorem 2.4.2 Let \( {v}_{1},\ldots ,{v}_{n} \in {R}^{n} \), all \( \left| {v}_{i}\right| \leq 1 \) . Let \( {p}_{1},\ldots ,{p}_{n} \in \left\lbrack {0,1}\right\rbrack \) be arbitrary and set \( w = {p}_{1}{v}_{1} + \ldots + {p}_{n}{v}_{n} \) . Then there exist \( {\epsilon }_{1},\ldots ,{\epsilon }_{n} \in \{ 0,1\} \... | Proof. Pick \( {\epsilon }_{i} \) independently with\n\n\[ \Pr \left\lbrack {{\epsilon }_{i} = 1}\right\rbrack = {p}_{i},\;\Pr \left\lbrack {{\epsilon }_{i} = 0}\right\rbrack = 1 - {p}_{i} \]\n\nThe random choice of \( {\epsilon }_{i} \) gives a random \( v \) and a random variable\n\n\[ X = {\left| w - v\right| }^{2} ... | Yes |
Theorem 2.5.1 Let \( {a}_{ij} = \pm 1 \) for \( 1 \leq i, j \leq n \) . Then there exist \( {x}_{i},{y}_{j} = \pm 1 \) , \( 1 \leq i, j \leq n \) so that \[ \mathop{\sum }\limits_{{i = 1}}^{n}\mathop{\sum }\limits_{{j = 1}}^{n}{a}_{ij}{x}_{i}{y}_{j} \geq \left( {\sqrt{\frac{2}{\pi }} + o\left( 1\right) }\right) {n}^{3/... | Proof.[Theorem 2.5.1] Forget the \( x \) ’s. Let \( {y}_{1},\ldots ,{y}_{n} = \pm 1 \) be selected independently and uniformly and set \[ {R}_{i} = \mathop{\sum }\limits_{{j = 1}}^{n}{a}_{ij}{y}_{j} \] \[ R = \mathop{\sum }\limits_{{i = 1}}^{n}\left| {R}_{i}\right| \] Fix \( i \) . Regardless of \( {a}_{ij},{a}_{ij}{y}... | Yes |
Theorem 1 [Brégman's Theorem]\n\n\\[ \n\\operatorname{per}\\left( A\\right) \\leq \\mathop{\\prod }\\limits_{{1 \\leq i \\leq n}}{\\left( {r}_{i}!\\right) }^{1/{r}_{i}} \n\\] | Claim 1 \\( \\operatorname{per}\\left( A\\right) \\leq G\\left\\lbrack L\\right\\rbrack \\) Proof.\n\nWe show this for any fixed \\( \\tau \\) . Set \\( ▩ = 1 \\) for convenience of notation. We use induction on the size of the matrix. Reorder, for convenience, so that the first row has ones in the first \\( r \\) colu... | Yes |
\[ {\left( \mathop{\prod }\limits_{{j = 1}}^{r}{t}_{j}^{{t}_{j}}\right) }^{1/r} \geq {t}^{t} \] | Proof. Taking logarithms this is equivalent to \[ \frac{1}{r}\mathop{\sum }\limits_{{j = 1}}^{r}{t}_{j}\ln {t}_{j} \geq t\ln t \] which follows from the convexity of the function \( f\left( x\right) = x\ln x \) . ∎ | Yes |
Theorem 3.1.1 For any integer \( n \)\n\n\[ R\left( {k, k}\right) > n - \left( \begin{array}{l} n \\ k \end{array}\right) {2}^{1 - \left( \begin{array}{l} k \\ 2 \end{array}\right) } \] | Proof. Consider a random two-coloring of the edges of \( {K}_{n} \) obtained by coloring each edge independently either red or blue, where each color is equally likely. For any set \( R \) of \( k \) vertices let \( {X}_{R} \) be the indicator random variable for the event that the induced subgraph of \( {K}_{n} \) on ... | Yes |
Theorem 3.1.3 For all integers \( n \) and \( p \in \left\lbrack {0,1}\right\rbrack \)\n\n\[ R\left( {k, l}\right) > n - \left( \begin{array}{l} n \\ k \end{array}\right) {p}^{\left( \begin{array}{l} k \\ 2 \end{array}\right) } - \left( \begin{array}{l} n \\ l \end{array}\right) {\left( 1 - p\right) }^{\left( \begin{ar... | Proof. In both cases we consider a random two-coloring of \( {K}_{n} \) obtained by coloring each edge independently either red or blue, where each edge is red with probability \( p \) . Let \( X \) be the number of red \( k \) -sets plus the number of blue \( l \) -sets. Linearity of Expectation gives\n\n\[ E\left\lbr... | No |
Theorem 3.2.1 Let \( G = \left( {V, E}\right) \) have \( n \) vertices and \( {nd}/2 \) edges, \( d \geq 1 \) . Then \( \alpha \left( G\right) \geq n/{2d} \) | Proof. Let \( S \subseteq V \) be a random subset defined by\n\n\[ \Pr \left\lbrack {v \in S}\right\rbrack = p \]\n\n\( p \) to be determined, the events \( v \in S \) being mutually independent. Let \( X = \left| S\right| \) and let \( Y \) be the number of edges in \( {\left. G\right| }_{S} \) . For each \( e = \{ i,... | Yes |
Theorem 3.3.1 There is a set \( S \) of \( n \) points in the unit square \( U \) such that \( T\left( S\right) \geq \) \( 1/\left( {{100}{n}^{2}}\right) \) . | ## Proof.\n\nWe first make a calculation. Let \( P, Q, R \) be independently and uniformly selected from \( U \) and let \( \mu = \mu \left( {PQR}\right) \) denote the area of the triangle \( {PQR} \) . We bound \( \Pr \left\lbrack {\mu \leq \epsilon }\right\rbrack \) as follows. Let \( x \) be the distance from \( P \... | Yes |
Theorem 3.4.1 Let \( C \) be bounded, convex, and centrally symmetric around the origin. Then\n\n\[ \delta \left( C\right) \geq {2}^{-d - 1} \] | Proof. Let \( P, Q \) be selected independently and uniformly from \( B\left( x\right) \) and consider the event \( \left( {C + P}\right) \cap \left( {C + Q}\right) \neq \varnothing \) . For this to occur we must have, for some \( {c}_{1},{c}_{2} \in C \)\n\n\[ P - Q = {c}_{1} - {c}_{2} = 2\frac{{c}_{1} - {c}_{2}}{2} \... | Yes |
Corollary 3.5. \( {2m}\left( n\right) = \Omega \left( {{2}^{n}{\left( n/\ln n\right) }^{1/2}}\right) \) | Proof. Bound \( 1 - p \leq {e}^{-p} \) . The function \( k{e}^{-{pn}} + {k}^{2}p \) is minimized at \( p = \) \( \ln \left( {n/k}\right) /n \) . Substituting back in, if\n\n\[ \frac{{k}^{2}}{n}\left\lbrack {1 + \ln \left( {n/k}\right) }\right\rbrack < 1 \]\n\nthen the condition of Theorem 3.5.1 holds. This inequality i... | Yes |
Lemma 3.5.3 \( E\left\lbrack {{\left( 1 + p\right) }^{i}{\left( 1 - p\right) }^{j}}\right\rbrack \leq 1 \) . | Proof. Fix a matching between \( e - \{ v\} \) and \( f - \{ v\} \), think of Mr. &Mrs. Jones; Mr. & Mrs. Smith, etc. Condition on how many of each pair (two Joneses, one Smith, no Taylors,..) come before \( v \) . The conditional expectation of \( {\left( 1 + p\right) }^{i}{\left( 1 - p\right) }^{j} \) splits into fac... | Yes |
Corollary 3.6.2 Under the assumptions of the theorem there exists a packing \( P \) of size \( \sim N/\left( {k + 1}\right) \) . | Proof. We have defined a random process which gives a packing with expected size \( \sim N/\left( {k + 1}\right) \) and our usual magic implies such a \( P \) must exist. - | No |
Theorem 1 [Erdős (1959)] For all \( k, l \) there exists a graph \( G \) with \( \operatorname{girth}\left( G\right) > l \) and \( \chi \left( G\right) > k \) . | Proof. Fix \( \theta < 1/l \) and let \( G \sim G\left( {n, p}\right) \) with \( p = {n}^{\theta - 1} \) . (I.e., \( G \) is a random graph on \( n \) vertices chosen by picking each pair of vertices as an edge randomly and independently with probability \( p \) ). Let \( X \) be the number of cycles of size at most \(... | Yes |
Theorem 4.1.1 [Chebyschev’s Inequality] For any positive \( \lambda \)\n\n\[ \Pr \left\lbrack {\left| {X - \mu }\right| \geq {\lambda \sigma }}\right\rbrack \leq \frac{1}{{\lambda }^{2}} \] | Proof.\n\n\[ {\sigma }^{2} = \operatorname{Var}\left\lbrack X\right\rbrack = E\left\lbrack {\left( X - \mu \right) }^{2}\right\rbrack \geq {\lambda }^{2}{\sigma }^{2}\Pr \left\lbrack {\left| {X - \mu }\right| \geq {\lambda \sigma }}\right\rbrack . \] | Yes |
Theorem 4.3.1\n\n\[ \Pr \left\lbrack {X = 0}\right\rbrack \leq \frac{\operatorname{Var}\left\lbrack X\right\rbrack }{E{\left\lbrack X\right\rbrack }^{2}} \] | Proof. Set \( \lambda = \mu /\sigma \) in Chebyschev’s Inequality. Then\n\n\[ \Pr \left\lbrack {X = 0}\right\rbrack \leq \Pr \left\lbrack {\left| {X - \mu }\right| \geq {\lambda \sigma }}\right\rbrack \leq \frac{1}{{\lambda }^{2}} = \frac{{\sigma }^{2}}{{\mu }^{2}}. \] | Yes |
Theorem 4.4.1 The property \( \omega \left( G\right) \geq 4 \) has threshold function \( {n}^{-2/3} \) . | Proof. For every 4-set \( S \) of vertices in \( G\left( {n, p}\right) \) let \( {A}_{S} \) be the event \ | No |
Theorem 4.4.2 Let \( H \) be a balanced graph with \( v \) vertices and e edges. Let \( A\left( G\right) \) be the event that \( H \) is a subgraph (not necessarily induced) of \( G \) . Then \( p = {n}^{-v/e} \) is the threshold function for \( A \) . | Proof. We follow the argument of Theorem 4.4.1. For each \( v \) -set \( S \) let \( {A}_{S} \) be the event that \( {\left. G\right| }_{S} \) contains \( H \) as a subgraph. Then\n\n\[ \n{p}^{e} \leq \Pr \left\lbrack {A}_{S}\right\rbrack \leq v!{p}^{e} \n\]\n\n(Any particular placement of \( H \) has probability \( {p... | Yes |
Theorem 4.4.3 In the notation of Theorem 4.4.2 if \( H \) is not balanced then \( p = {n}^{-v/e} \) is not the threshold function for \( A \) . | Proof. Let \( {H}_{1} \) be a subgraph of \( H \) with \( {v}_{1} \) vertices, \( {e}_{1} \) edges and \( {e}_{1}/{v}_{1} > e/v \) . Let \( \alpha \) satisfy \( v/e < \alpha < {v}_{1}/{e}_{1} \) and set \( p = {n}^{-\alpha } \) . The expected number of copies of \( {H}_{1} \) is then \( o\left( 1\right) \) so almost al... | Yes |
Theorem 4.4.4 Let \( H \) be strictly balanced with \( v \) vertices, \( e \) edges and a automor-phisms. Let \( X \) be the number of copies of \( H \) in \( G\left( {n, p}\right) \) . Assume \( p > > {n}^{-v/e} \) . Then almost always | Proof. Label the vertices of \( H \) by \( 1,\ldots, v \) . For each ordered \( {x}_{1},\ldots ,{x}_{v} \) let \( {A}_{{x}_{1},\ldots ,{x}_{v}} \) be the event that \( {x}_{1},\ldots ,{x}_{v} \) provides a copy of \( H \) in that order. Specifically we define \[ {A}_{{x}_{1},\ldots ,{x}_{v}} : \{ i, j\} \in E\left( H\r... | Yes |
Theorem 4.4.5 Let \( H \) be any fixed graph. For every subgraph \( {H}^{\prime } \) of \( H \) (including \( H \) itself) let \( {X}_{{H}^{\prime }} \) denote the number of copies of \( {H}^{\prime } \) in \( G\left( {n, p}\right) \) . Assume \( p \) is such that \( E\left\lbrack {X}_{{H}^{\prime }}\right\rbrack \righ... | Proof. Let \( H \) have \( v \) vertices and \( e \) edges. As in Theorem 4.4.4 it suffices to show \( {\Delta }^{ * } = o\left( {E\left\lbrack X\right\rbrack }\right) \) . We split \( {\Delta }^{ * } \) into a finite number of terms. For each \( {H}^{\prime } \) with \( w \) vertices and \( f \) edges we have those \(... | Yes |
Theorem 4.5.1 Let \( k = k\left( n\right) \) satisfy \( k \sim 2{\log }_{2}n \) and \( f\left( k\right) \rightarrow \infty \) . Then almost always \( \omega \left( G\right) \geq k \) . | Proof. For each \( k \) -set \( S \) let \( {A}_{S} \) be the event \ | No |
Theorem 4.7.1 For every integer \( r \geq 2 \) and reals \( k \geq 1 \) and \( a > 0 \), there are \( \gamma = \gamma \left( {r, k, a}\right) > 0 \) and \( {d}_{0} = {d}_{0}\left( {r, k, a}\right) \) such that for every \( n \geq D \geq {d}_{0} \) the following holds.\n\nEvery \( r \) -uniform hypergraph \( H = \left( ... | The basic idea in the proof is simple. Fixing a small \( \epsilon > 0 \) one shows that a random set of roughly \( {\epsilon n}/r \) edges has, with high probability, only some \( O\left( {{\epsilon }^{2}n}\right) \) vertices covered more than once, and hence covers at least \( {\epsilon n} - O\left( {{\epsilon }^{2}n}... | Yes |
Lemma 4.7.2 For every integer \( r \geq 2 \) and reals \( K \geq 1 \) and \( \epsilon > 0 \), and for every real \( {\delta }^{\prime } > 0 \), there are \( \delta = \delta \left( {r, K,\epsilon ,{\delta }^{\prime }}\right) > 0 \) and \( {D}_{0} = {D}_{0}\left( {r, K,\epsilon ,{\delta }^{\prime }}\right) \) such that f... | Proof. Throughout the proof we assume, whenever this is needed, that \( D \) (and hence \( n \) ) are sufficiently large. We denote by \( {\delta }_{1},{\delta }_{2},\cdots \) positive constants (that can be explicitly estimated) that tend to 0 when \( \delta \) tends to 0 and \( D \) tends to infinity (for fixed \( r,... | Yes |
Theorem 4.7.3 (Rödl) For \( k, l \) fixed,\n\n\[ M\left( {n, k, l}\right) \leq \left( {1 + o\left( 1\right) }\right) \frac{\left( \begin{matrix} n \\ l \end{matrix}\right) }{\left( \begin{matrix} k \\ l \end{matrix}\right) }\n\]\n\nwhere the \( o\left( 1\right) \) term tends to zero as \( n \) tends to infinity. | Proof. Put \( r = \left( \begin{array}{l} k \\ l \end{array}\right) \) and let \( H \) be the \( r \) -uniform hypergraph whose vertices are all \( l \) -subsets of \( \{ 1,2,\ldots, n\} \), and whose edges are all collections of \( \left( \begin{array}{l} k \\ l \end{array}\right) l \) -tuples that lie in a \( k \) -s... | Yes |
Theorem 1 There exists a positive constant \( c \) such that for every \( n \)\n\n\[ P\left( n\right) \leq c{n}^{3/2}\frac{n!}{{2}^{n - 1}} \] | Proof. For a tournament \( T \), denote by \( P\left( T\right) \) the number of directed Hamilton paths of \( T \) . Similarly, \( C\left( T\right) \) denotes the number of directed Hamilton cycles of \( T \), and \( F\left( T\right) \) denotes the number of spanning subgraphs of \( T \) in which the indegree and the o... | Yes |
For every two integers \( a, b \) satisfying \( b \geq a + 2 > a \geq 1 \) the inequality\n\n\[{\left( a!\right) }^{1/a} \cdot {\left( b!\right) }^{1/b} < {\left( \left( a + 1\right) !\right) }^{1/\left( {a + 1}\right) } \cdot {\left( \left( b - 1\right) !\right) }^{1/\left( {b - 1}\right) }\]\n\nholds. | Proof. The assertion is simply that \( f\left( a\right) < f\left( {b - 1}\right) \), for the function \( f \) defined by \( f\left( a\right) = {\left( a!\right) }^{1/a}/{\left( \left( a + 1\right) !\right) }^{1/\left( {a + 1}\right) } \) . Thus, it suffices to show that for every integer \( x \geq 2, f\left( {x - 1}\ri... | Yes |
Corollary 3 Define \( g\left( x\right) = {\left( x!\right) }^{1/x} \) . For every integer \( S \geq n \) the maximum of the function \( \mathop{\prod }\limits_{{i = 1}}^{n}g\left( {x}_{i}\right) \) subject to the constraints \( \mathop{\sum }\limits_{{i = 1}}^{n}{x}_{i} = S \) and \( {x}_{i} \geq 1 \) are integers, is ... | Proof. If there are two indices \( i \) and \( j \) such that \( {x}_{i} \geq {x}_{j} + 2 \) then, by Lemma 2, the value of the product would increase once we add one to \( {x}_{j} \) and subtract one from \( {x}_{i} \) . \( ▱ \) | Yes |
Corollary 5.1.2 [The Local Lemma; Symmetric Case] Let \( {A}_{1},{A}_{2},\ldots ,{A}_{n} \) be events in an arbitrary probability space. Suppose that each event \( {A}_{i} \) is mutually independent of a set of all the other events \( {A}_{j} \) but at most \( d \), and that \( \Pr \left( {A}_{i}\right) \leq p \) for a... | Proof. If \( d = 0 \) the result is trivial. Otherwise, by the assumption there is a dependency digraph \( D = \left( {V, E}\right) \) for the events \( {A}_{1},\ldots ,{A}_{n} \) in which for each \( i,\{ j \) : \( \left( {i, j}\right) \in E\} \mid \leq d \) . The result now follows from Lemma 5.1.1 by taking \( {x}_{... | Yes |
Theorem 5.2.1 Let \( H = \left( {V, E}\right) \) be a hypergraph in which every edge has at least \( k \) elements, and suppose that each edge of \( H \) intersects at most \( d \) other edges. If \( e\left( {d + 1}\right) \leq {2}^{k - 1} \) then \( H \) has property \( B \) . | Proof. Color each vertex \( v \) of \( H \), randomly and independently, either blue or red (with equal probability). For each edge \( f \in E \), let \( {A}_{f} \) be the event that \( f \) is monochromatic. Clearly \( \Pr \left( {A}_{f}\right) = 2/{2}^{\left| f\right| } \leq 1/{2}^{k - 1} \) . Moreover, each event \(... | Yes |
Theorem 5.2.2 Let \( m \) and \( k \) be two positive integers satisfying\n\n\[ e\left( {m\left( {m - 1}\right) + 1}\right) k{\left( 1 - \frac{1}{k}\right) }^{m} \leq 1 \]\n\n(5.7)\n\nThen, for any set \( S \) of \( m \) real numbers there is a \( k \) -coloring so that each translation \( x + S \) (for \( x \in \mathb... | Proof. We first fix a finite subset \( X \subseteq \mathbb{R} \) and show the existence of a \( k \) -coloring so that each translation \( x + S \) (for \( x \in X \) ) is multicolored. This is an easy consequence of the Local Lemma. Indeed, put \( Y = \mathop{\bigcup }\limits_{{x \in X}}\left( {x + S}\right) \) and le... | Yes |
Proposition 5.3.1 If \( e\left( {\left( \begin{array}{l} k \\ 2 \end{array}\right) \left( \begin{matrix} n \\ k - 2 \end{matrix}\right) + 1}\right) \cdot {2}^{1 - \left( \begin{array}{l} k \\ 2 \end{array}\right) } < 1 \) then \( R\left( {k, k}\right) > n \) . | A short computation shows that this gives \( R\left( {k, k}\right) > \frac{\sqrt{2}}{e}\left( {1 + o\left( 1\right) }\right) k{2}^{k/2} \) , only a factor 2 improvement on the bound obtained by the straightforward probabilistic method. | Yes |
Theorem 5.4.1 Let \( \mathcal{F} = {\left\{ {B}_{i}\right\} }_{i \in I} \) be a \( k \) -fold covering of the 3 dimensional Euclidean space by open unit balls. Suppose, further, than no point of \( {\mathbb{R}}^{3} \) is contained in more than \( t \) members of \( \mathcal{F} \) . If\n\n\[ e \cdot {t}^{3}{2}^{18}/{2}^... | Proof. Define an infinite hypergraph \( H = \left( {V\left( H\right), E\left( H\right) }\right) \) as follows. The set of vertices of \( H, V\left( H\right) \), is simply \( \mathcal{F} = {\left\{ {B}_{i}\right\} }_{i \in I} \) . For each \( x \in {\mathbb{R}}^{3} \) let \( {E}_{x} \) be the set of balls \( {B}_{i} \in... | Yes |
Proposition 5.5.3 Let \( H = \left( {V, E}\right) \) be a graph with maximum degree \( d \), and let \( V = {V}_{1} \cup {V}_{2} \cup \cdots \cup {V}_{r} \) be a partition of \( V \) into \( r \) pairwise disjoint sets. Suppose each set \( {V}_{i} \) is of cardinality \( \left| {V}_{i}\right| \geq {2ed} \), where \( e ... | Proof. Clearly we may assume that each set \( {V}_{i} \) is of cardinality precisely \( g = \lceil {2ed}\rceil \) (otherwise, simply replace each \( {V}_{i} \) by a subset of cardinality \( g \) of it, and replace \( H \) by its induced subgraph on the union of these \( r \) new sets). Let us pick from each set \( {V}_... | Yes |
Theorem 5.5.4 Let \( G = \\left( {U, F}\\right) \) be a \( d \) -regular digraph with directed girth \( g \\geq {8ed} \) . Then\n\n\[ \n\\operatorname{dla}\\left( G\\right) = d + 1 \n\] | Proof. As is well known, \( F \) can be partitioned into \( d \) pairwise disjoint 1-regular spanning subgraphs \( {F}_{1},\\ldots ,{F}_{d} \) of \( G \) . (This is an easy consequence of the Hall-König Theorem; let \( H \) be the bipartite graph whose two classes of vertices \( A \) and \( B \) are copies of \( U \), ... | Yes |
Lemma 5.5.5 Let \( G = \left( {V, E}\right) \) be a d-regular directed graph, where \( d \) is sufficiently large, and let \( p \) be an integer satisfying \( {10}\sqrt{d} \leq p \leq {20}\sqrt{d} \) . Then, there is a \( p \) - coloring of the vertices of \( G \) by the colors \( 0,1,2,\ldots, p - 1 \) with the follow... | Proof. Let \( f : V \rightarrow \{ 0,1,\ldots, p - 1\} \) be a random vertex coloring of \( V \) by \( p \) colors, where for each \( v \in V, f\left( v\right) \in \{ 0,1,\ldots, p - 1\} \) is chosen according to a uniform distribution. For every vertex \( v \in V \) and every color \( i,0 \leq i < p \), let \( {A}_{v,... | Yes |
Theorem 5.5.6 There is an absolute constant \( c > 0 \) such that for every \( d \) -regular digrapn \( G \)\n\n\[ \mathrm{d}\operatorname{la}\left( G\right) \leq d + c{d}^{3/4}{\left( \log d\right) }^{1/2}. \] | We note that by being a little more careful, we can improve the error term to \( {c}^{\prime }{d}^{2/3}{\left( \log d\right) }^{1/3} \) . Since the edges of any undirected \( d = {2f} \) -regular graph can be oriented so that the resulting digraph is \( f \) -regular, and since any \( \left( {{2f} - 1}\right) \) -regul... | No |
Theorem 5.7.1 Let \( n, d \) be such that, setting \( D = d{\left( d - 1\right) }^{3} \) there exists a decomposition \( n = {n}_{1} + {n}_{2} + {n}_{3} \) with\n\n\[ \n{16D}\left( {1 + d}\right) < {2}^{{n}_{1}} \]\n\n\[ \n{16D}\left( {1 + d}\right) < {2}^{{n}_{2}} \]\n\n\[ \n{2e}\left( {1 + d}\right) < {2}^{{n}_{3}}. ... | Proof. The First Pass. During this pass points will be either Red, Blue, uncolored or saved. We move through the points \( j \in \Omega \) sequentially, coloring them Red or Blue at random, flipping a fair coin. After each \( j \) is colored we check all \( {A}_{i} \ni j \) . If \( {A}_{i} \) now has \( {n}_{1} \) poin... | Yes |
Theorem 1 [Alon and Linial (1989) ] If \( e\left( {{\Delta \delta } + 1}\right) {\left( 1 - \frac{1}{k}\right) }^{\delta } < 1 \) then \( D \) contains a (directed, simple) cycle of length \( 0\left( {\;\operatorname{mod}\;k}\right) \) . | Proof. Clearly we may assume that every outdegree is precisely \( \delta \), since otherwise we can consider a subgraph of \( D \) with this property.\n\nLet \( f : V \rightarrow \{ 0,1,\ldots, k - 1\} \) be a random coloring of \( V \), obtained by choosing, for each \( v \in V, f\left( v\right) \in \{ 0,\ldots, k - 1... | Yes |
Corollary 6.1.4 Let \( X \) be a family of subsets of a finite set \( N \) and define\n\n\[ \nX \smallsetminus X = \\left\\{ {F \smallsetminus {F}^{\\prime } : F,{F}^{\\prime } \\in X}\\right\\} .\n\]\n\nThen \( \\left| {X \smallsetminus X}\\right| \\geq \\left| X\\right| \) . | Proof.\n\nLet \( L \) be the distributive lattice of all subsets of \( N \) . By applying Corollary 6.1.3 to \( X \) and \( Y = \\{ N \\smallsetminus F : F \\in X\\} \) we obtain:\n\n\[ \n{\\left| X\\right| }^{2} = \\left| X\\right| \\cdot \\left| Y\\right| \\leq \\left| {X \\cup Y}\\right| \\cdot \\left| {X \\cap Y}\\... | Yes |
Theorem 6.2.1 [The \( {FKG} \) inequality]\n\nLet \( L \) be a finite distributive lattice and let \( \mu : L \rightarrow {\mathbb{R}}^{ + } \) be a log-supermodular function. Then, for any two increasing functions \( f, g : L \rightarrow {\mathbb{R}}^{ + } \) we have\n\n\[ \left( {\mathop{\sum }\limits_{{x \in L}}\mu ... | Proof.\n\nDefine four functions \( \alpha ,\beta ,\gamma ,\delta : L \rightarrow {\mathbb{R}}^{ + } \) as follows. For each \( x \in L \)\n\n\[ \alpha \left( x\right) \; = \mu \left( x\right) f\left( x\right) ,\;\beta \left( x\right) = \mu \left( x\right) g\left( x\right) \]\n\n\[ \gamma \left( x\right) = \mu \left( x\... | Yes |
Proposition 6.3.1 Let \( \mathcal{A} \) and \( \mathcal{B} \) be two monotone increasing families of subsets of \( N = \{ 1,2,\ldots, n\} \) and let \( \mathcal{C} \) and \( \mathcal{D} \) be two monotone decreasing families of subsets of \( N \) . Then\n\n\[ \Pr \left( {\mathcal{A} \cap \mathcal{B}}\right) \geq \Pr \l... | Proof. Let \( f : P\left( N\right) \rightarrow {\mathbb{R}}^{ + } \) be the characteristic function of \( \mathcal{A} \), i.e., \( f\left( A\right) = 0 \) if \( A \notin \mathcal{A} \) and \( f\left( A\right) = 1 \) if \( A \in \mathcal{A} \) . Similarly, let \( g \) be the characteristic function of\n\n\( B \) . By th... | Yes |
\[ \alpha \left( G\right) \geq \mathop{\sum }\limits_{{v \in V}}\frac{1}{{d}_{v} + 1} \] | Proof. Let \( < \) be a uniformly chosen total ordering of \( V \) . Define\n\n\[ I = \{ v \in V : \{ v, w\} \in E \Rightarrow v < w\} \]\n\nLet \( {X}_{v} \) be the indicator random variable for \( v \in I \) and \( X = \mathop{\sum }\limits_{{v \in V}}{X}_{v} = \left| I\right| \) . For each \( v \)\n\n\[ E\left\lbrac... | Yes |
Theorem 2 [Turán (1941) ]Turán’s Theorem Let \( H \) have \( n \) vertices and \( e \) edges. Then \( \alpha \left( H\right) \geq m \) and \( \alpha \left( H\right) = m \Leftrightarrow H \cong {G}_{n, e} \) . | Proof. \( {G}_{n, e} \) has \( \mathop{\sum }\limits_{{v \in V}}{\left( {d}_{v} + 1\right) }^{-1} = m \) since each clique contributes 1 to the sum. Fixing \( e = \mathop{\sum }\limits_{{v \in V}}{d}_{v}/2,\mathop{\sum }\limits_{{v \in V}}{\left( {d}_{v} + 1\right) }^{-1} \) is minimized with the \( {d}_{v} \) as close... | Yes |
Theorem 7.2.1 [Azuma’s Ineqality] Let \( 0 = {X}_{0},\ldots ,{X}_{m} \) be a martingale with\n\n\[ \left| {{X}_{i + 1} - {X}_{i}}\right| \leq 1 \]\n\nfor all \( 0 \leq i < m \) . Let \( \lambda > 0 \) be arbitrary. Then\n\n\[ \Pr \left\lbrack {{X}_{m} > \lambda \sqrt{m}}\right\rbrack < {e}^{-{\lambda }^{2}/2}. \] | Proof. Set, with foresight, \( \alpha = \lambda /\sqrt{m} \) . Set \( {Y}_{i} = {X}_{i} - {X}_{i - 1} \) so that \( \left| {Y}_{i}\right| \leq 1 \) and \( E\left\lbrack {{Y}_{i} \mid {X}_{i - 1},{X}_{i - 2},\ldots ,{X}_{0}}\right\rbrack = 0 \) . Then, as in A.1.16,\n\n\[ E\left\lbrack {{e}^{\alpha {Y}_{i}} \mid {X}_{i ... | Yes |
Theorem 7.2.3 When \( f \) satisfies the edge Lipschitz condition, the corresponding edge exposure martingale satisfies \( \left| {{X}_{i + 1} - {X}_{i}}\right| \leq 1 \) . When \( f \) satisfies the vertex Lipschitz condition the corresponding vertex exposure martingale satisfies \( \mid {X}_{i + 1} - \) \( {X}_{i} \m... | We prove these results in a more general context later. They have the intuitive sense that if knowledge of a particular vertex or edge cannot change \( f \) by more than one then exposing a vertex or edge should not change the expectation of \( f \) by more than one. | No |
Theorem 7.2.4 [Shamir and Spencer (1987) ] Let \( n, p \) be arbitrary and let \( c = \) \( E\left\lbrack {\chi \left( G\right) }\right\rbrack \) where \( G \sim G\left( {n, p}\right) \) . Then \[ \Pr \left\lbrack {\left| {\chi \left( G\right) - c}\right| > \lambda \sqrt{n - 1}}\right\rbrack < 2{e}^{-{\lambda }^{2}/2}.... | Proof. Consider the vertex exposure martingale \( {X}_{1},\ldots ,{X}_{n} \) on \( G\left( {n, p}\right) \) with \( f\left( G\right) = \) \( \chi \left( G\right) \) . A single vertex can always be given a new color so the vertex Lipschitz condition applies. Now apply Azuma's Inequality in the form of Corollary 7.2.2. - | Yes |
Lemma 7.3.1 \( E\left\lbrack Y\right\rbrack \geq \frac{{n}^{2}}{2{k}^{4}}\left( {1 + o\left( 1\right) }\right) \) . | Proof. Let \( \mathcal{K} \) denote the family of \( k \) -cliques of \( G \) so that \( f\left( k\right) = \mu = E\left\lbrack \left| \mathcal{K}\right| \right\rbrack \) . Let \( W \) denote the number of unordered pairs \( \{ A, B\} \) of \( k \) -cliques of \( G \) with \( 2 \leq \left| {A \cap B}\right| < k \) . Th... | Yes |
Theorem 7.3.2\n\n\[ \Pr \left\lbrack {\omega \left( G\right) < k}\right\rbrack < {e}^{-\left( {c + o\left( 1\right) }\right) \frac{{n}^{2}}{{\ln }^{8}n}} \] | Proof. Let \( {Y}_{0},\ldots ,{Y}_{m}, m = \left( \begin{array}{l} n \\ 2 \end{array}\right) \), be the edge exposure martingale on \( G\left( {n,1/2}\right) \) with the function \( Y \) just defined. The function \( Y \) satisfies the edge Lipschitz condition\nas adding a single edge can only add at most one clique to... | Yes |
Lemma 7.3.4 Let \( \alpha, c \) be fixed \( \alpha > \frac{5}{6} \) . Let \( p = {n}^{-\alpha } \) . Then almost always every \( c\sqrt{n} \) vertices of \( G = G\left( {n, p}\right) \) may be \( 3 \) -colored. | Proof. If not, let \( T \) be a minimal set which is not 3-colorable. As \( T - \{ x\} \) is 3-colorable, \( x \) must have internal degree at least 3 in \( T \) for all \( x \in T \) . Thus if \( T \) has \( t \) vertices it must have at least \( \frac{3t}{2} \) edges. The probability of this occuring for some \( T \)... | Yes |
Theorem 7.4.1 Let \( L \) satisfy the Lipschitz condition. Then the corresponding martingale satisfies\n\n\[ \left| {{X}_{i + 1}\left( h\right) - {X}_{i}\left( h\right) }\right| \leq 1 \]\n\nfor all \( 0 \leq i < m, h \in {A}^{B} \) . | Proof. Let \( H \) be the family of \( {h}^{\prime } \) which agree with \( h \) on \( {B}_{i + 1} \) . Then\n\n\[ {X}_{i + 1}\left( h\right) = \mathop{\sum }\limits_{{{h}^{\prime } \in H}}L\left( {h}^{\prime }\right) {w}_{{h}^{\prime }} \]\n\nwhere \( {w}_{{h}^{\prime }} \) is the conditional probability that \( g = {... | Yes |
Theorem 7.4.3 For all \( \epsilon > 0 \) there exists \( \delta > 0 \) so that the following holds. Suppose Paul has a strategy for finding \( Y \) such that every line of questioning has total variance at most \( {\sigma }^{2} \) . Then \[ \Pr \left\lbrack {\left| {Y - E\left\lbrack Y\right\rbrack }\right| > {\alpha \... | Proof. For simplicity we replace \( Y \) by \( Y - E\left\lbrack Y\right\rbrack \) so that we shall henceforth assume \( E\left\lbrack Y\right\rbrack = 0 \) . By symmetry we shall bound only the upper tail of \( Y \) . We set, with foresight, \( \lambda = \alpha /\left\lbrack {\sigma \left( {1 + \epsilon }\right) }\rig... | Yes |
Theorem 7.5.2\n\n\[ \Pr \left\lbrack {X - E\left\lbrack X\right\rbrack > \lambda \sqrt{n}}\right\rbrack < {e}^{-{\lambda }^{2}/2} \]\n\n\[ \Pr \left\lbrack {X - E\left\lbrack X\right\rbrack < - \lambda \sqrt{n}}\right\rbrack < {e}^{-{\lambda }^{2}/2}. \] | Proof. Consider \( \{ - 1, + 1{\} }^{n} \) as the underlying probability space with all \( \left( {{\epsilon }_{1},\ldots ,{\epsilon }_{n}}\right) \) equally likely. Then \( X \) is a random variable and we define a martingale \( {X}_{0},\ldots ,{X}_{n} = \) \( X \) by exposing one \( {\epsilon }_{i} \) at a time. The ... | Yes |
Theorem 7.5.3 Let \( \epsilon ,\lambda > 0 \) satisfy \( {e}^{-{\lambda }^{2}/2} = \epsilon \) . Then\n\n\[ \left| A\right| \geq \epsilon {2}^{n} \Rightarrow \left| {B\left( {A,{2\lambda }\sqrt{n}}\right) }\right| \geq \left( {1 - \epsilon }\right) {2}^{n}. \] | Proof. Consider \( \{ 0,1{\} }^{n} \) as the underlying probability space, all points equally likely. For \( y \in \{ 0,1{\} }^{n} \) set\n\n\[ X\left( y\right) = \mathop{\min }\limits_{{x \in A}}\rho \left( {x, y}\right) \]\n\nLet \( {X}_{0},{X}_{1},\ldots ,{X}_{n} = X \) be the martingale given by exposing one coordi... | Yes |
\[ \rho \left( {A,\overrightarrow{x}}\right) = \mathop{\min }\limits_{{\overrightarrow{v} \in V\left( {A,\overrightarrow{x}}\right) }}\left| \overrightarrow{v}\right| \] | Proof. Let \( \overrightarrow{v} \in V\left( {A,\overrightarrow{x}}\right) \) achieve this minimum. The hyperplane through \( \overrightarrow{v} \) perpendicular to the line from the origin to \( \overrightarrow{v} \) then separates \( V\left( {A,\overrightarrow{x}}\right) \) from the origin so that all \( \overrightar... | Yes |
Theorem 7.6.2\n\n\[ \int_{\Omega} \exp \left[ \frac{1}{4} \rho^2 \left( A, \overrightarrow{x} \right) \right] d\overrightarrow{x} \leq \frac{1}{\Pr \left[ A \right] }. \] | Proof.[Theorem 7.6.2] We use induction on the dimension \( n \). For \( n = 1, \rho \left( A, \overrightarrow{x} \right) = 1 \) if \( \overrightarrow{x} \notin A \), zero otherwise so that\n\n\[ \int \exp \left[ \frac{1}{4} \rho^2 \left( A, \overrightarrow{x} \right) \right] = \Pr \left[ A \right] + \left( 1 - \Pr \lef... | Yes |
Theorem 7.7.1 Under the above assumptions and for all \( b, t \)\n\n\[ \Pr \left\lbrack {X \leq b - t\sqrt{f\left( b\right) }}\right\rbrack \Pr \left\lbrack {X \geq b}\right\rbrack \leq {e}^{-{t}^{2}/4}. \] | Proof. Set \( A = \{ x : h\left( x\right) < b - t\sqrt{f\left( b\right) }\} \) . Now suppose \( h\left( y\right) \geq b \) . We claim \( y \notin {A}_{t} \) . Let \( I \) be a set of indices of size at most \( f\left( b\right) \) that certifies \( h\left( y\right) \geq b \) as given above. Define \( {\alpha }_{i} = 0 \... | Yes |
Theorem 1 [Weierstrass Approximation Theorem] For every continuous real function \( f : \left\lbrack {0,1}\right\rbrack \mapsto R \) and every \( \epsilon > 0 \), there is a polynomial \( p\left( x\right) \) such that \( \left| {p\left( x\right) - f\left( x\right) }\right| \leq \epsilon \) for all \( x \in \left\lbrack... | Proof. Since a continuous \( f : \left\lbrack {0,1}\right\rbrack \mapsto R \) is uniformly continuous there is a \( \delta > 0 \) such that if \( x,{x}^{\prime } \in \left\lbrack {0,1}\right\rbrack \) and \( \left| {x - {x}^{\prime }}\right| \leq \delta \) then \( \left| {f\left( x\right) - f\left( {x}^{\prime }\right)... | Yes |
Theorem 8.1.1 [The Janson Inequality] Let \( {B}_{i}, i \in I \) , \( \Delta, M,\mu \) be as above and assume all \( \Pr \left\lbrack {B}_{i}\right\rbrack \leq \epsilon \) . Then\n\n\[ M \leq \Pr \left\lbrack {{ \land }_{i \in I}\overline{{B}_{i}}}\right\rbrack \leq M{e}^{\frac{1}{1 - \epsilon }\frac{\Delta }{2}} \]\n\... | For each \( i \in I \)\n\n\[ \Pr \left\lbrack \overline{{B}_{i}}\right\rbrack = 1 - \Pr \left\lbrack {B}_{i}\right\rbrack \leq {e}^{-\Pr \left\lbrack {B}_{i}\right\rbrack } \]\n\nso, multiplying over \( i \in I \) ,\n\n\[ M \leq {e}^{-\mu } \] | No |
Theorem 8.1.2 (when it applies) often gives a much stronger result than Chebyschev’s Inequality as used in Chapter 4. In \( §{4.3} \) we saw \( \operatorname{Var}\left\lbrack X\right\rbrack \leq \mu + \Delta \) so that \[ \Pr \left\lbrack {{ \land }_{i \in I}\overline{{B}_{i}}}\right\rbrack = \Pr \left\lbrack {X = 0}\r... | Proof.[Theorem 8.1.1] The lower bound follows immediately. Order the index set \( I = \{ 1,\ldots, m\} \) for convenience. For \( 1 \leq i \leq m \) \[ \Pr \left\lbrack {{B}_{i} \mid { \land }_{1 \leq j < i}\overline{{B}_{j}}}\right\rbrack \leq \Pr \left\lbrack {B}_{i}\right\rbrack \] so \[ \Pr \left\lbrack {\overline{... | Yes |
Theorem 8.3.1 Suppose there is a constant \( \mu \) so that\n\n\[ E\left\lbrack X\right\rbrack = {S}^{\left( 1\right) } \rightarrow \mu \]\n\nand such that for every fixed \( r \)\n\n\[ E\left\lbrack {{X}^{\left( r\right) }/r!}\right\rbrack = {S}^{\left( r\right) } \rightarrow {\mu }^{r}/r! \]\n\nThen\n\n\[ \Pr \left\l... | Proof. We do only the case \( t = 0 \) . Fix \( \epsilon > 0 \) . Choose \( s \) so that\n\n\[ \left| {\mathop{\sum }\limits_{{r = 0}}^{{2s}}{\left( -1\right) }^{r}\frac{{\mu }^{r}}{r!} - {e}^{-\mu }}\right| \leq \frac{\epsilon }{2} \]\n\nThe Bonferroni Inequalities state that, in general, the inclusion-exclusion formu... | No |
Lemma 8.4.1 With the above notation and for any integer \( s \)\n\n\[ \Pr \left\lbrack {\text{ there exists a disfam }J,\left| J\right| = s}\right\rbrack \leq \frac{{\mu }^{s}}{s!}. \] | Proof. Let \( \mathop{\sum }\limits^{ * } \) denote the sum over all \( s \) -sets \( J \subseteq I \) with no \( j \sim {j}^{\prime } \) . Let \( \mathop{\sum }\limits^{o} \) denote the sum over ordered \( s \) -tuples \( \left( {{j}_{1},\ldots ,{j}_{s}}\right) \) with \( \left\{ {{j}_{1},\ldots ,{j}_{s}}\right\} \) f... | Yes |
Lemma 8.4.2 With the above notation and for any integer \( s \)\n\n\[\n\\begin{array}{ll} \\Pr \\left\\lbrack {\\text{ there exists a maxdisfam }J,\\left| J\\right| = s}\\right\\rbrack & \\leq \\frac{{\\mu }^{s}}{s!}{e}^{-{\\mu }_{s}}{e}^{\\frac{\\Delta }{2}} \\\\\n & \\leq \\frac{{\\mu }^{s}}{s!}{e}^{-\\mu }{e}^{s\\nu... | Proof. As in Lemma 8.4.1 we bound this probability by \( \\mathop{\\sum }\\limits^{ * } \) of \( J = \\left\\{ {{j}_{1},\\ldots ,{j}_{s}}\\right\\} \) being a maxdisfam. For this to occur \( J \) must first be a disfam and then \( { \\land }^{ * }\\overline{{B}_{i}} \), where \( { \\land }^{ * } \) is the conjunction o... | Yes |
Theorem 8.5.1 Set \( p = \frac{\ln n}{n}\omega \left( n\right) \) where \( \omega \left( n\right) \rightarrow \infty \) arbitrarily slowly. Then in \( G\left( {n, p}\right) \) almost always\n\n\[ \deg \left( x\right) \sim \left( {n - 1}\right) p \]\n\nfor all vertices \( x \) . | This is actually a large deviation result. It suffices to show the following. | No |
Theorem 8.5.2 Set \( p = \frac{\ln n}{n}\omega \left( n\right) \) where \( \omega \left( n\right) \rightarrow \infty \) arbitrarily slowly. Let \( x \in G \) be fixed. Fix \( \epsilon > 0 \) . Then\n\n\[ \Pr \left\lbrack {\left| {\deg \left( x\right) - \left( {n - 1}\right) p}\right| > \epsilon \left( {n - 1}\right) p}... | Proof. As \( \deg \left( x\right) \sim B\left( {n - 1, p}\right) \), i.e., it is a Binomial random variable with the above parameters, we have from A.1.14 that\n\n\[ \Pr \left\lbrack {\left| {\deg \left( x\right) - \left( {n - 1}\right) p}\right| > \epsilon \left( {n - 1}\right) p}\right\rbrack < 2{e}^{-{c}_{\epsilon }... | Yes |
Theorem 8.5.4 Let \( p \) be such that \( \mu > > \ln n \) . Let \( x \in G \) be fixed. Fix \( \epsilon > 0 \) Then\n\n\[ \Pr \left\lbrack {\left| {N\left( x\right) - \mu }\right| > {\epsilon \mu }}\right\rbrack = o\left( {n}^{-1}\right) . \] | Proof. We shall prove this under the further assumption \( p = {n}^{-2/3 + o\left( 1\right) } \) (or, equivalently, \( \mu = {n}^{o\left( 1\right) } \) ) which could be removed by technical methods. We now have, in the notation of Lemmas 8.4.1,8.4.2 \( {\nu \mu },\Delta = o\left( 1\right) \) . Let \( P \) denote the Po... | Yes |
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