import OAI.Combinatorics.CycleDecomposition.ExpansionBoxes namespace OAI universe u universe cycleUniverse1 cycleUniverse2 cycleUniverse3 cycleUniverse4 cycleUniverse5 cycleUniverse6 cycleUniverse7 section open Filter Asymptotics Real open scoped Topology noncomputable section open MeasureTheory ProbabilityTheory Finset section namespace ErdosGallai.Scale open Finset SimpleGraph Classical variable {V : Type cycleUniverse1} [Fintype V] [DecidableEq V] lemma edgeCount_eq_card {V : Type cycleUniverse2} [_contextInstance1 : Fintype V] [_contextInstance2 : DecidableEq V] (G : SimpleGraph V) (S : Finset (Sym2 V)) (hS : ∀ e, e ∈ S ↔ e ∈ G.edgeSet) : edgeCount G = S.card := by unfold edgeCount congr 1 ext e simp only [SimpleGraph.mem_edgeFinset] exact (hS e).symm def IsCycleGraph (K : SimpleGraph V) : Prop := ∃ a : V, ∃ p : K.Walk a a, p.IsCycle ∧ K.edgeSet = p.edgeSet def NoLongCycle (G : SimpleGraph V) (ℓ : ℝ) : Prop := ∀ a : V, ∀ p : G.Walk a a, p.IsCycle → (p.length : ℝ) < ℓ lemma NoLongCycle.mono {V : Type cycleUniverse3} [_contextInstance1 : Fintype V] [_contextInstance2 : DecidableEq V] {G H : SimpleGraph V} {ℓ : ℝ} (h : NoLongCycle G ℓ) (hle : H ≤ G) : NoLongCycle H ℓ := by intro a p hp simpa using h a (p.mapLe hle) (hp.mapLe hle) lemma cycle_graph_of_walk (G : SimpleGraph V) {a : V} (p : G.Walk a a) (hp : p.IsCycle) : ∃ K : SimpleGraph V, K ≤ G ∧ IsCycleGraph K ∧ edgeCount K = p.length := by classical let K := SimpleGraph.fromEdgeSet p.edgeSet have he : K.edgeSet = p.edgeSet := by rw [SimpleGraph.edgeSet_fromEdgeSet,sdiff_eq_left] exact Set.disjoint_left.mpr (fun e he hd => G.not_isDiag_of_mem_edgeSet (p.edges_subset_edgeSet he) hd) have hle : K ≤ G := SimpleGraph.edgeSet_subset_edgeSet.mp (he ▸ p.edges_subset_edgeSet) have htransfer : ∀ e ∈ p.edges, e ∈ K.edgeSet := by intro e he'; rw [he]; exact he' refine ⟨K,hle,⟨a,p.transfer K htransfer,hp.transfer htransfer,?_⟩,?_⟩ · simpa using he · rw [edgeCount_eq_card K p.edges.toFinset (by intro e; simp only [List.mem_toFinset,he]; rfl)] exact (List.toFinset_card_of_nodup hp.isTrail.edges_nodup).trans p.length_edges lemma edgeCount_sdiff_add {G K : SimpleGraph V} (hle : K ≤ G) : edgeCount (G \ K) + edgeCount K = edgeCount G := by classical rw [edgeCount_eq_card (G \ K) (G.edgeFinset \ K.edgeFinset) (by intro e; simp only [Finset.mem_sdiff,SimpleGraph.mem_edgeFinset,SimpleGraph.edgeSet_sdiff, Set.mem_sdiff]), edgeCount_eq_card K K.edgeFinset (by intro e; simp), edgeCount_eq_card G G.edgeFinset (by intro e; simp)] exact Finset.card_sdiff_add_card_eq_card (SimpleGraph.edgeFinset_mono hle) lemma edgeCount_cycle_pos {K : SimpleGraph V} (h : IsCycleGraph K) : 0 < edgeCount K := by classical obtain ⟨a,p,hp,he⟩ := h have hf : K.edgeFinset = p.edges.toFinset := by ext e simp only [SimpleGraph.mem_edgeFinset,he,List.mem_toFinset] rfl simp only [edgeCount,hf,List.toFinset_card_of_nodup hp.isTrail.edges_nodup, SimpleGraph.Walk.length_edges] exact hp.three_le_length.trans_lt' (by omega) theorem remove_long_cycles (G : SimpleGraph V) (ℓ : ℝ) : ∃ (H : SimpleGraph V) (L : List (SimpleGraph V)), H ≤ G ∧ NoLongCycle H ℓ ∧ (∀ K ∈ L, IsCycleGraph K ∧ ℓ ≤ (edgeCount K : ℝ)) ∧ GraphPartition G (H::L) ∧ (L.length : ℝ)*ℓ + edgeCount H ≤ edgeCount G := by classical induction hn : edgeCount G using Nat.strong_induction_on generalizing G with | h n ih => by_cases hstop : NoLongCycle G ℓ · refine ⟨G,[],le_rfl,hstop,by simp,?_,?_⟩ · intro x y; simp · simp [← hn] · obtain ⟨a,p,hp,hplen⟩ : ∃ a, ∃ p : G.Walk a a, p.IsCycle ∧ ℓ ≤ (p.length : ℝ) := by by_contra hh apply hstop intro a p hp by_contra hlen exact hh ⟨a,p,hp,le_of_not_gt hlen⟩ obtain ⟨K,hle,hK,hKlen⟩ := cycle_graph_of_walk G p hp have heq := edgeCount_sdiff_add hle have hpos := edgeCount_cycle_pos hK have hlt : edgeCount (G \ K) < n := by omega obtain ⟨H,L,hH,hNL,hL,hpart,hbudget⟩ := ih _ hlt (G \ K) rfl refine ⟨H,K::L,hH.trans sdiff_le,hNL,?_,?_,?_⟩ · intro J hJ rcases List.mem_cons.mp hJ with rfl | hJ · exact ⟨hK,by simpa [hKlen] using hplen⟩ · exact hL J hJ · intro x y have hh := @hH x y have hk := @hle x y have he := hpart x y simp only [edgeOccurrences_cons,SimpleGraph.sdiff_adj] at he ⊢ by_cases hG : G.Adj x y <;> by_cases hK : K.Adj x y <;> by_cases hH : H.Adj x y <;> simp_all · have heqR : (edgeCount (G \ K) : ℝ) + edgeCount K = edgeCount G := by exact_mod_cast heq have hKbound : ℓ ≤ (edgeCount K : ℝ) := by simpa [hKlen] using hplen simp only [List.length_cons,Nat.cast_add,Nat.cast_one] rw [← hn] nlinarith open Real Filter Asymptotics lemma longCycle_power_margin : ∀ᶠ u : ℝ in atTop, 4 ≤ u ∧ u^(199/200:ℝ) ≤ ((1/32:ℝ)/(logb 2 u)^2)^2*u/100 := by have hlog := (isLittleO_log_rpow_rpow_atTop (4:ℝ) (s:=1/200) (by norm_num)).const_mul_left ((102400:ℝ)/(log 2)^4) filter_upwards [eventually_ge_atTop (4:ℝ),hlog.eventuallyLE] with u hu hh have hu0 : 0 < u := by linarith have hln : 0 ≤ log u := log_nonneg (by linarith) have hln2 : 0 < log 2 := log_pos (by norm_num) have hLu : 0 < logb 2 u := logb_pos (by norm_num) (by linarith) have hr0 : 0 ≤ u^(1/200:ℝ) := rpow_nonneg hu0.le _ have hnum0 : 0 ≤ (102400:ℝ)/(log 2)^4 * (log u)^(4:ℝ) := by positivity simp only [Real.norm_eq_abs,abs_of_nonneg hnum0,abs_of_nonneg hr0] at hh have hden : (102400:ℝ)*(logb 2 u)^4 ≤ u^(1/200:ℝ) := by calc (102400:ℝ)*(logb 2 u)^4 = (102400:ℝ)/(log 2)^(4:ℕ) * (log u)^(4:ℕ) := by simp only [Real.logb,div_pow]; ring _ ≤ u^(1/200:ℝ) := by simpa only [show (4:ℝ) = ((4:ℕ):ℝ) by norm_num, Real.rpow_natCast] using hh have hm := mul_le_mul_of_nonneg_left hden (rpow_nonneg hu0.le (199/200:ℝ)) have he : u^(199/200:ℝ)*u^(1/200:ℝ) = u := by rw [← Real.rpow_add hu0]; norm_num rw [he] at hm refine ⟨hu,?_⟩ have hid : ((1/32:ℝ)/(logb 2 u)^2)^2*u/100 = u/(102400*(logb 2 u)^4) := by field_simp ; ring rw [hid] exact (le_div_iff₀ (by positivity : (0:ℝ) < 102400*(logb 2 u)^4)).mpr (by nlinarith) lemma longCycle_scale_ready : ∃ Dsp : ℝ, ∀ D ≥ Dsp, 4 ≤ D ∧ 4 ≤ D^(51/50:ℝ) ∧ ∀ u : ℝ, D^(51/50:ℝ) < u → 0 < (1/32:ℝ)/(logb 2 u)^2 ∧ (1/32:ℝ)/(logb 2 u)^2 ≤ 1 ∧ 1 ≤ ((1/32:ℝ)/(logb 2 u)^2)^2*u/100 ∧ D^(101/100:ℝ) < ((1/32:ℝ)/(logb 2 u)^2)^2*u/100 := by obtain ⟨Q,hQ⟩ := eventually_atTop.mp longCycle_power_margin refine ⟨max 4 Q,?_⟩ intro D hD have hD4 : 4 ≤ D := (le_max_left _ _).trans hD have hD1 : 1 ≤ D := by linarith have hDp : 0 < D := by linarith have hTD : D ≤ D^(51/50:ℝ) := by simpa only [Real.rpow_one] using Real.rpow_le_rpow_of_exponent_le hD1 (by norm_num : (1:ℝ) ≤ 51/50) refine ⟨hD4,hD4.trans hTD,?_⟩ intro u hu have huQ : Q ≤ u := (le_max_right 4 Q).trans (hD.trans (hTD.trans hu.le)) obtain ⟨hu4,hq⟩ := hQ u huQ have hlog : 1 ≤ logb 2 u := by have h := Real.logb_le_logb_of_le (b:=2) (x:=2) (y:=u) (by norm_num) (by norm_num) (by linarith) norm_num [Real.logb] at h exact h have hd : 0 < (logb 2 u)^2 := by positivity have hα : 0 < (1/32:ℝ)/(logb 2 u)^2 := by positivity have hα1 : (1/32:ℝ)/(logb 2 u)^2 ≤ 1 := by apply (div_le_one hd).mpr nlinarith [sq_nonneg (logb 2 u-1)] have hmargin : D^(101/100:ℝ) < u^(199/200:ℝ) := by have hstrict := Real.rpow_lt_rpow (Real.rpow_nonneg hDp.le (51/50:ℝ)) hu (by norm_num : (0:ℝ) < 199/200) rw [← Real.rpow_mul hDp.le] at hstrict have hle := Real.rpow_le_rpow_of_exponent_le hD1 (by norm_num : (101/100:ℝ) ≤ (51/50)*(199/200)) exact hle.trans_lt hstrict have hpow1 : 1 ≤ D^(101/100:ℝ) := Real.one_le_rpow hD1 (by norm_num) exact ⟨hα,hα1,hpow1.trans (hmargin.le.trans hq),hmargin.trans_le hq⟩ lemma external_induce_image (G : SimpleGraph V) (A : Finset V) (hG : SupportedOn G A) (U : Finset {x // x ∈ A}) : externalOn G (U.image Subtype.val) = (externalOn (G.induce (A : Set V)) U).image Subtype.val := by classical ext y constructor · intro hy obtain ⟨_,hyU,x,hx,hxy⟩ := Finset.mem_filter.mp hy have hyA := (hG hxy).2 obtain ⟨x',hx',heq⟩ := Finset.mem_image.mp hx refine Finset.mem_image.mpr ⟨⟨y,hyA⟩,Finset.mem_filter.mpr ⟨Finset.mem_univ _,?_,?_⟩,rfl⟩ · intro hy' exact hyU (Finset.mem_image.mpr ⟨⟨y,hyA⟩,hy',rfl⟩) · exact ⟨x',hx',by simpa only [SimpleGraph.induce_adj,heq] using hxy⟩ · intro hy obtain ⟨y',hy',rfl⟩ := Finset.mem_image.mp hy obtain ⟨_,hyU,x,hx,hxy⟩ := Finset.mem_filter.mp hy' refine Finset.mem_filter.mpr ⟨Finset.mem_univ _,?_,?_⟩ · intro hy obtain ⟨z,hz,heq⟩ := Finset.mem_image.mp hy have heq' : z = y' := Subtype.ext heq exact hyU (heq' ▸ hz) · exact ⟨x,Finset.mem_image.mpr ⟨x,hx,rfl⟩,hxy⟩ lemma sparse_neighborhood_from_cycle_exclusion (G : SimpleGraph V) (A : Finset V) (hG : SupportedOn G A) (ℓ : ℝ) (hNL : NoLongCycle G ℓ) (hα : 0 < (1/32:ℝ)/(logb 2 A.card)^2) (hα1 : (1/32:ℝ)/(logb 2 A.card)^2 ≤ 1) (hq : 1 ≤ ((1/32:ℝ)/(logb 2 A.card)^2)^2*A.card/100) (hmargin : ℓ < ((1/32:ℝ)/(logb 2 A.card)^2)^2*A.card/100) : ∃ U ⊆ A, U.Nonempty ∧ 2*U.card ≤ A.card ∧ ((externalOn G U).card : ℝ) < (1/32:ℝ)*U.card/(logb 2 A.card)^2 := by classical by_contra hno let H := G.induce (A : Set V) let α := (1/32:ℝ)/(logb 2 A.card)^2 have hexp : VertexExpansion H α := by intro U hU hhalf let W := U.image Subtype.val have hW : W ⊆ A := by intro x hx obtain ⟨x',hx',rfl⟩ := Finset.mem_image.mp hx exact x'.property have hcard : W.card = U.card := Finset.card_image_of_injective _ Subtype.val_injective have hWne : W.Nonempty := hU.image _ have hWhalf : 2*W.card ≤ A.card := by simpa [hcard] using hhalf have hs : ¬ ((externalOn G W).card : ℝ) < (1/32:ℝ)*W.card/(logb 2 A.card)^2 := fun hs => hno ⟨W,hW,hWne,hWhalf,hs⟩ have hbound := le_of_not_gt hs have heq := external_induce_image G A hG U have hcardext := congrArg Finset.card heq rw [Finset.card_image_of_injective _ Subtype.val_injective] at hcardext have hext : externalNeighbors H U = externalOn H U := by rfl rw [hext] change α*(U.card:ℝ) ≤ (externalOn H U).card rw [← hcardext] dsimp [α] rw [← hcard] calc (1/32:ℝ)/(logb 2 A.card)^2 * (W.card:ℝ) = (1/32:ℝ)*(W.card:ℝ)/(logb 2 A.card)^2 := by ring _ ≤ ((externalOn G W).card:ℝ) := hbound have hq' : 1 ≤ α^2*(Fintype.card {x // x ∈ A})/100 := by simpa [α] using hq obtain ⟨a,p,hp,hplen⟩ := long_cycle_of_vertex_expansion H α hα hα1 hexp hq' let f : H →g G := ⟨Subtype.val,fun h => h⟩ have hc := hp.map (f:=f) (by exact Subtype.val_injective) have hlen := hNL a.val (p.map f) hc simp only [SimpleGraph.Walk.length_map] at hlen have hplen' : α^2*(A.card:ℝ)/100 < (p.length:ℝ) := by simpa using hplen dsimp [α] at hplen' linarith lemma splitPotential_scale_bound (D x : ℝ) (hD : 4 ≤ D) : splitPotential (D^(51/50:ℝ)) x ≤ 1+2/logb 2 D := by have hD1 : 1 ≤ D := by linarith have hDp : 0 < D := by linarith have hLD : 2 ≤ logb 2 D := by have h := Real.logb_le_logb_of_le (b:=2) (x:=4) (y:=D) (by norm_num) (by norm_num) hD have h4 : logb 2 (4:ℝ) = 2 := by rw [show (4:ℝ) = 2^(2:ℕ) by norm_num,Real.logb_pow] norm_num rw [h4] at h exact h have hLD0 : 0 < logb 2 D := by linarith have hlogT : logb 2 (D^(51/50:ℝ)/2) = (51/50:ℝ)*logb 2 D-1 := by rw [Real.logb_div (Real.rpow_pos_of_pos hDp _).ne' (by norm_num), Real.logb_rpow_eq_mul_logb_of_pos hDp] norm_num have hT0 : 0 < logb 2 (D^(51/50:ℝ)/2) := by rw [hlogT]; linarith have hrecip : 1/logb 2 (D^(51/50:ℝ)/2) ≤ 2/logb 2 D := by apply (div_le_div_iff₀ hT0 hLD0).mpr rw [hlogT] linarith have hbasepos : 0 < D^(51/50:ℝ)/2 := by positivity have hmaxlog := Real.logb_le_logb_of_le (b:=2) (by norm_num) hbasepos (le_max_right x (D^(51/50:ℝ)/2)) have hmax0 : 0 < logb 2 (max x (D^(51/50:ℝ)/2)) := lt_of_lt_of_le hT0 hmaxlog have hinv := one_div_pos.mpr hmax0 dsimp [splitPotential] linarith lemma partition_replace_head {G H : SimpleGraph V} {C L : List (SimpleGraph V)} (hG : GraphPartition G (H::C)) (hH : GraphPartition H L) : GraphPartition G (C++L) := by intro x y have h := hG x y simp only [edgeOccurrences_cons] at h rw [edgeOccurrences_append,hH x y] omega theorem scale_boxes : ∃ Dsp : ℝ, ∀ D ≥ Dsp, ∀ (V : Type u) [Fintype V] [DecidableEq V] (G : SimpleGraph V) (A : Finset V), A.Nonempty → SupportedOn G A → 2*(edgeCount G : ℝ)/A.card ≤ D → ∃ (C : List (SimpleGraph V)) (L : List (AssignedPiece V)), (∀ K ∈ C, IsCycleGraph K ∧ D^(101/100:ℝ) ≤ (edgeCount K : ℝ)) ∧ (C.length : ℝ) ≤ A.card/(2*D^(1/100:ℝ)) ∧ (∀ P ∈ L, P.vertices ⊆ A ∧ P.vertices.Nonempty ∧ (P.vertices.card : ℝ) ≤ D^(51/50:ℝ)) ∧ (∀ x ∈ A, ∃ P ∈ L, x ∈ P.vertices) ∧ GraphPartition G (C++L.map AssignedPiece.graph) ∧ (L.map (fun P => (P.vertices.card : ℝ))).sum ≤ (1+2/logb 2 D)*A.card := by classical obtain ⟨Dsp,hsp⟩ := longCycle_scale_ready refine ⟨Dsp,?_⟩ intro D hD V _ _ G A hA hG hdeg obtain ⟨hD4,hT4,hnum⟩ := hsp D hD have hDp : 0 < D := by linarith have hA0 : (0:ℝ) < A.card := by exact_mod_cast hA.card_pos obtain ⟨H,C,hHG,hNL,hC,hpartC,hbudget⟩ := remove_long_cycles G (D^(101/100:ℝ)) have hHA : SupportedOn H A := fun _ _ h => hG (hHG h) have hsplit : ∀ B ⊆ A, ∀ K : SimpleGraph V, K ≤ H → SupportedOn K B → D^(51/50:ℝ) < (B.card:ℝ) → ∃ U ⊆ B, U.Nonempty ∧ 2*U.card ≤ B.card ∧ ((externalOn K U).card : ℝ) < (1/32:ℝ)*U.card/(logb 2 B.card)^2 := by intro B hB K hK hs hlarge obtain ⟨ha,ha1,hq,hm⟩ := hnum B.card hlarge exact sparse_neighborhood_from_cycle_exclusion K B hs _ (hNL.mono hK) ha ha1 hq hm obtain ⟨L,hL,hcov,hpartL,hcost⟩ := box_recursion H A (D^(51/50:ℝ)) hT4 hA hHA hsplit refine ⟨C,L,hC,?_,hL,hcov,partition_replace_head hpartC hpartL,?_⟩ · have hdeg' := (div_le_iff₀ hA0).mp hdeg have he : D^(101/100:ℝ) = D*D^(1/100:ℝ) := by rw [show (101/100:ℝ) = 1+1/100 by norm_num,Real.rpow_add hDp,Real.rpow_one] rw [he] at hbudget apply (le_div_iff₀ (by positivity : (0:ℝ) < 2*D^(1/100:ℝ))).mpr have hnonneg : (0:ℝ) ≤ edgeCount H := Nat.cast_nonneg _ nlinarith · have hfbound := splitPotential_scale_bound D A.card hD4 have hplain : (L.map (fun P => (P.vertices.card:ℝ))).sum ≤ (L.map (fun P => (P.vertices.card:ℝ)*splitPotential (D^(51/50:ℝ)) P.vertices.card)).sum := by apply List.sum_le_sum intro P hP have hb := (splitPotential_bounds (D^(51/50:ℝ)) P.vertices.card hT4).1 have hm := mul_le_mul_of_nonneg_left hb (Nat.cast_nonneg P.vertices.card) simpa using hm have hmul := mul_le_mul_of_nonneg_left hfbound hA0.le nlinarith end ErdosGallai.Scale namespace ErdosGallai.Scale open Finset SimpleGraph Classical Real variable {V : Type cycleUniverse4} [Fintype V] [DecidableEq V] lemma edgeCount_sup_le (G H : SimpleGraph V) : edgeCount (G ⊔ H) ≤ edgeCount G + edgeCount H := by classical simpa [edgeCount, SimpleGraph.edgeFinset_sup] using Finset.card_union_le G.edgeFinset H.edgeFinset lemma crossing_count (G : SimpleGraph V) (A B : Finset V) (hd : Disjoint A B) : edgeCount (crossingGraph G A B) = cutCount G A B := by classical have hinj : Set.InjOn (fun e : V × V => s(e.1,e.2)) (G.interedges A B : Set (V×V)) := by intro e he f hf hh rcases Sym2.mk_eq_mk_iff.mp hh with hh | hh · exact hh · have heA := (G.mem_interedges_iff.mp he).1 have hfB := (G.mem_interedges_iff.mp hf).2.1 have heq : e.1 = f.2 := congrArg Prod.fst hh exact False.elim (Finset.disjoint_left.mp hd heA (heq.symm ▸ hfB)) have heq : (crossingGraph G A B).edgeFinset = (G.interedges A B).image (fun e : V×V => s(e.1,e.2)) := by ext e induction e using Sym2.inductionOn with | _ x y => simp only [SimpleGraph.mem_edgeFinset, SimpleGraph.mem_edgeSet] constructor · rintro ⟨hxy,(⟨hx,hy⟩|⟨hx,hy⟩)⟩ · exact Finset.mem_image.mpr ⟨(x,y),G.mem_interedges_iff.mpr ⟨hx,hy,hxy⟩,rfl⟩ · exact Finset.mem_image.mpr ⟨(y,x),G.mem_interedges_iff.mpr ⟨hy,hx,hxy.symm⟩, Sym2.eq_swap⟩ · rintro h obtain ⟨⟨a,b⟩,hab,he⟩ := Finset.mem_image.mp h obtain ⟨ha,hb,hg⟩ := G.mem_interedges_iff.mp hab rcases (Sym2.mk_eq_mk_iff (p := (a,b)) (q := (x,y))).mp he with he | he · cases he; exact ⟨hg,Or.inl ⟨ha,hb⟩⟩ · cases he; exact ⟨hg.symm,Or.inr ⟨hb,ha⟩⟩ simp only [edgeCount,heq,Finset.card_image_of_injOn hinj,cutCount] lemma log2_split_budget {a b : ℕ} (ha : 0 < a) (hb : 0 < b) : min a b + a*a.log2 + b*b.log2 ≤ (a+b)*(a+b).log2 := by have hm : Monotone Nat.log2 := by intro m n h simpa only [Nat.log2_eq_log_two] using (Nat.log_mono_right (b:=2) h) have hla := hm (show a ≤ a+b by omega) have hlb := hm (show b ≤ a+b by omega) rcases le_total a b with hab | hba · have hhalf := hm (show 2*a ≤ a+b by omega) rw [Nat.log2_two_mul (by omega)] at hhalf rw [min_eq_left hab] nlinarith · have hhalf := hm (show 2*b ≤ a+b by omega) rw [Nat.log2_two_mul (by omega)] at hhalf rw [min_eq_right hba] nlinarith lemma partition_combine {G K G₁ G₂ H₁ H₂ : SimpleGraph V} {L₁ L₂ : List (SimpleGraph V)} (hp₁ : GraphPartition G₁ (H₁::L₁)) (hp₂ : GraphPartition G₂ (H₂::L₂)) (hcover : ∀ x y, G.Adj x y ↔ K.Adj x y ∨ G₁.Adj x y ∨ G₂.Adj x y) (hd₁ : Disjoint K G₁) (hd₂ : Disjoint K G₂) (hd₃ : Disjoint G₁ G₂) : GraphPartition G ((K ⊔ H₁ ⊔ H₂)::(L₁++L₂)) := by classical have hle₁ : H₁ ≤ G₁ := partition_member_le hp₁ (by simp) have hle₂ : H₂ ≤ G₂ := partition_member_le hp₂ (by simp) intro x y have h₁ := hp₁ x y have h₂ := hp₂ x y have hd₁ := SimpleGraph.disjoint_left.mp hd₁ x y have hd₂ := SimpleGraph.disjoint_left.mp hd₂ x y have hd₃ := SimpleGraph.disjoint_left.mp hd₃ x y have hh₁ := @hle₁ x y have hh₂ := @hle₂ x y have hc := hcover x y simp only [edgeOccurrences_cons,edgeOccurrences_append,SimpleGraph.sup_adj] at * by_cases hK : K.Adj x y <;> by_cases hG₁ : G₁.Adj x y <;> by_cases hG₂ : G₂.Adj x y <;> by_cases hH₁ : H₁.Adj x y <;> by_cases hH₂ : H₂.Adj x y <;> simp_all lemma cut_split_properties {V : Type cycleUniverse5} [_contextInstance1 : Fintype V] [_contextInstance2 : DecidableEq V] (G : SimpleGraph V) (A U : Finset V) (hG : SupportedOn G A) (_hU : U ⊆ A) : (∀ x y, G.Adj x y ↔ (crossingGraph G U (A\U)).Adj x y ∨ (restrictGraph G U).Adj x y ∨ (restrictGraph G (A\U)).Adj x y) ∧ Disjoint (crossingGraph G U (A\U)) (restrictGraph G U) ∧ Disjoint (crossingGraph G U (A\U)) (restrictGraph G (A\U)) ∧ Disjoint (restrictGraph G U) (restrictGraph G (A\U)) := by constructor · intro x y change G.Adj x y ↔ (G.Adj x y ∧ _) ∨ (G.Adj x y ∧ _) ∨ (G.Adj x y ∧ _) have hs := @hG x y simp only [Finset.mem_sdiff] tauto · simp only [SimpleGraph.disjoint_left, crossingGraph, restrictGraph,Finset.mem_sdiff] tauto lemma empty_supported_graph {V : Type cycleUniverse6} [_contextInstance1 : Fintype V] [_contextInstance2 : DecidableEq V] {G : SimpleGraph V} {A : Finset V} (hG : SupportedOn G A) (hA : A.card ≤ 1) : G = ⊥ := by ext x y constructor · intro h obtain ⟨hx,hy⟩ := hG h have hxy := Finset.card_le_one.mp hA x hx y hy exact False.elim (G.ne_of_adj h hxy) · simp theorem cut_extraction (G : SimpleGraph V) (A : Finset V) (hG : SupportedOn G A) (σ : ℝ) (hσ : 0 ≤ σ) : ∃ (R : List (AssignedPiece V)) (H : SimpleGraph V), (∀ P ∈ R, P.vertices ⊆ A ∧ 2 ≤ P.vertices.card ∧ CutExpansionOn P.graph P.vertices σ) ∧ R.Pairwise (fun P Q => Disjoint P.vertices Q.vertices) ∧ GraphPartition G (H :: R.map AssignedPiece.graph) ∧ SupportedOn H A ∧ (edgeCount H : ℝ) ≤ σ * A.card * A.card.log2 := by classical induction hcard : A.card using Nat.strong_induction_on generalizing A G with | h n ih => by_cases hsmall : A.card ≤ 1 · have he := empty_supported_graph hG hsmall refine ⟨[],⊥,by simp,by simp,?_,?_,?_⟩ · subst G intro x y simp · intro x y h; exact False.elim h · simp only [edgeCount_bot,Nat.cast_zero] positivity by_cases hexp : CutExpansionOn G A σ · refine ⟨[⟨A,G,hG⟩],⊥,?_,by simp,?_,?_,?_⟩ · intro P hP have he := List.mem_singleton.mp hP subst P exact ⟨fun _ h => h,by change 2 ≤ A.card; omega,hexp⟩ · intro x y; simp · intro x y h; exact False.elim h · simp only [edgeCount_bot,Nat.cast_zero] positivity unfold CutExpansionOn at hexp push Not at hexp obtain ⟨U,hU,hcut⟩ := hexp have hdisUB : Disjoint U (A\U) := Finset.disjoint_left.mpr (fun _ hu hb => (Finset.mem_sdiff.mp hb).2 hu) have hcnt : (0 : ℝ) ≤ cutCount G U (A\U) := Nat.cast_nonneg _ have hσpos : 0 < σ := by by_contra h have : σ = 0 := le_antisymm (le_of_not_gt h) hσ rw [this,zero_mul] at hcut linarith have hUpos : 0 < U.card := by by_contra h have hz : U.card = 0 := by omega simp [hz] at hcut linarith [min_le_left (0:ℝ) ((A\U).card:ℝ)] have hBpos : 0 < (A\U).card := by by_contra h have hz : (A\U).card = 0 := by omega simp [hz] at hcut linarith [min_le_right (U.card:ℝ) (0:ℝ)] have hcardSum := Finset.card_sdiff_add_card_eq_card hU have hUlt : U.card < n := by omega have hBlt : (A\U).card < n := by omega obtain ⟨R₁,H₁,hR₁,hd₁,hp₁,hs₁,hc₁⟩ := ih U.card hUlt (restrictGraph G U) U (restrictGraph_supported G U) rfl obtain ⟨R₂,H₂,hR₂,hd₂,hp₂,hs₂,hc₂⟩ := ih (A\U).card hBlt (restrictGraph G (A\U)) (A\U) (restrictGraph_supported G (A\U)) rfl let K := crossingGraph G U (A\U) obtain ⟨hcover,hdis₁,hdis₂,hdis₃⟩ := cut_split_properties G A U hG hU refine ⟨R₁++R₂,K ⊔ H₁ ⊔ H₂,?_,?_,?_,?_,?_⟩ · intro P hP rcases List.mem_append.mp hP with hP | hP · obtain ⟨hPA,hPc,hPe⟩ := hR₁ P hP exact ⟨hPA.trans hU,hPc,hPe⟩ · obtain ⟨hPA,hPc,hPe⟩ := hR₂ P hP exact ⟨hPA.trans Finset.sdiff_subset,hPc,hPe⟩ · apply List.pairwise_append.mpr refine ⟨hd₁,hd₂,?_⟩ intro P hP Q hQ exact hdisUB.mono (hR₁ P hP).1 (hR₂ Q hQ).1 · simpa only [List.map_append] using partition_combine hp₁ hp₂ hcover hdis₁ hdis₂ hdis₃ · intro x y h rcases h with (h|h)|h · exact hG h.1 · exact ⟨hU (hs₁ h).1,hU (hs₁ h).2⟩ · exact ⟨(Finset.mem_sdiff.mp (hs₂ h).1).1,(Finset.mem_sdiff.mp (hs₂ h).2).1⟩ · have hcardK : edgeCount K = cutCount G U (A\U) := crossing_count G U (A\U) hdisUB have hcost : (edgeCount (K⊔H₁⊔H₂) : ℝ) ≤ edgeCount K + edgeCount H₁ + edgeCount H₂ := by exact_mod_cast (edgeCount_sup_le (K⊔H₁) H₂).trans (Nat.add_le_add_right (edgeCount_sup_le K H₁) _) have hpot := log2_split_budget hUpos hBpos have hpotR : min (U.card:ℝ) ((A\U).card:ℝ) + U.card*U.card.log2 + (A\U).card*(A\U).card.log2 ≤ (A.card:ℝ) * A.card.log2 := by have he : U.card+(A\U).card = A.card := by omega rw [he] at hpot exact_mod_cast hpot have hpay := mul_le_mul_of_nonneg_left hpotR hσ rw [hcardK] at hcost rw [← hcard] nlinarith lemma cast_log2_le_logb {n : ℕ} (hn : 0 < n) : (n.log2 : ℝ) ≤ Real.logb 2 n := by apply (Real.le_logb_iff_rpow_le (by norm_num : (1:ℝ)<2) (by exact_mod_cast hn)).mpr rw [Real.rpow_natCast, Nat.log2_eq_log_two] exact_mod_cast Nat.pow_log_le_self 2 hn.ne' lemma expansion_order_lower {V : Type cycleUniverse7} [_contextInstance1 : Fintype V] [_contextInstance2 : DecidableEq V] {G : SimpleGraph V} {A : Finset V} {σ : ℝ} (hA : 2 ≤ A.card) (he : CutExpansionOn G A σ) : σ+1 ≤ A.card := by classical obtain ⟨x,hx⟩ := Finset.card_pos.mp (show 0 < A.card by omega) have hsub : {x} ⊆ A := Finset.singleton_subset_iff.mpr hx have hcard := Finset.card_sdiff_add_card_eq_card hsub simp only [Finset.card_singleton] at hcard have hB : (1:ℝ) ≤ (A\{x}).card := by exact_mod_cast (show 1 ≤ (A\{x}).card by omega) have hc := he {x} hsub simp only [Finset.card_singleton,Nat.cast_one,min_eq_left hB,mul_one] at hc have hbound : cutCount G {x} (A\{x}) ≤ (A\{x}).card := by simpa only [cutCount,Finset.card_singleton,one_mul] using G.card_interedges_le_mul {x} (A\{x}) have hboundR : (cutCount G {x} (A\{x}) : ℝ) ≤ (A\{x}).card := by exact_mod_cast hbound have heqR : ((A\{x}).card:ℝ)+1 = A.card := by exact_mod_cast hcard linarith open Filter Asymptotics Real in lemma cut_scale_ready : ∀ᶠ D : ℝ in atTop, 1 ≤ D ∧ 2*D^(9/10:ℝ)*logb 2 (D^(51/50:ℝ)) ≤ D^(19/20:ℝ) := by have hlog2 : 0 < log 2 := log_pos (by norm_num) have he := ((isLittleO_log_rpow_atTop (by norm_num : (0:ℝ)<1/20)).const_mul_left ((51/25:ℝ)/log 2)).eventuallyLE filter_upwards [eventually_ge_atTop (1:ℝ),he] with D hD h have hDp : 0 < D := by linarith have hn : 0 ≤ (51/25:ℝ)/log 2 * log D := mul_nonneg (by positivity) (Real.log_nonneg hD) have hh : (51/25:ℝ)/log 2 * log D ≤ D^(1/20:ℝ) := by simpa only [Real.norm_eq_abs,abs_of_nonneg hn,abs_of_nonneg (Real.rpow_nonneg hDp.le _)] using h refine ⟨hD,?_⟩ calc 2*D^(9/10:ℝ)*logb 2 (D^(51/50:ℝ)) = D^(9/10:ℝ)*((51/25:ℝ)/log 2*log D) := by rw [Real.logb_rpow_eq_mul_logb_of_pos hDp] simp only [Real.logb] ring _ ≤ D^(9/10:ℝ)*D^(1/20:ℝ) := mul_le_mul_of_nonneg_left hh (by positivity) _ = D^(19/20:ℝ) := by rw [← Real.rpow_add hDp]; norm_num theorem scale_cut_extraction : ∃ Dcut : ℝ, ∀ D ≥ Dcut, ∀ (V : Type u) [Fintype V] [DecidableEq V] (G : SimpleGraph V) (A : Finset V), A.Nonempty → SupportedOn G A → (A.card : ℝ) ≤ D^(51/50:ℝ) → ∃ (R : List (AssignedPiece V)) (H : SimpleGraph V), (∀ P ∈ R, P.vertices ⊆ A ∧ D^(9/10:ℝ) ≤ P.vertices.card ∧ (P.vertices.card:ℝ) ≤ D^(51/50:ℝ) ∧ CutExpansionOn P.graph P.vertices (D^(9/10:ℝ))) ∧ R.Pairwise (fun P Q => Disjoint P.vertices Q.vertices) ∧ GraphPartition G (H :: R.map AssignedPiece.graph) ∧ SupportedOn H A ∧ 2*(edgeCount H : ℝ)/A.card ≤ D^(19/20:ℝ) := by classical obtain ⟨Dcut,hready⟩ := Filter.eventually_atTop.mp cut_scale_ready refine ⟨Dcut,?_⟩ intro D hD V _ _ G A hA hG hhi obtain ⟨hD₁,hscale⟩ := hready D hD have hDp : 0 < D := by linarith have hAp : 0 < A.card := Finset.card_pos.mpr hA have hAr : (0:ℝ) < A.card := by exact_mod_cast hAp obtain ⟨R,H,hR,hd,hp,hs,hcost⟩ := cut_extraction G A hG (D^(9/10:ℝ)) (by positivity) refine ⟨R,H,?_,hd,hp,hs,?_⟩ · intro P hP obtain ⟨hsub,hsize,hexp⟩ := hR P hP have hlo := expansion_order_lower hsize hexp have hle : (P.vertices.card:ℝ) ≤ A.card := by exact_mod_cast Finset.card_le_card hsub exact ⟨hsub,by linarith,hle.trans hhi,hexp⟩ · have hlog := (cast_log2_le_logb hAp).trans (Real.logb_le_logb_of_le (by norm_num : (1:ℝ)<2) hAr hhi) have hmul := mul_le_mul_of_nonneg_left hlog (show 0 ≤ D^(9/10:ℝ)*(A.card:ℝ) by positivity) have hscaleA := mul_le_mul_of_nonneg_right hscale hAr.le apply (div_le_iff₀ hAr).mpr nlinarith end ErdosGallai.Scale end end end end OAI