| import OAI.Combinatorics.ListHadwiger.RecursiveInduction |
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| namespace OAI |
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| namespace LinearListHadwiger.Recursion |
| open Finset Stage SmallList SimpleGraph |
| attribute [local instance] Classical.propDecidable |
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| theorem final_bound_or_obstruction : ∃ C : ℕ, 1≤C ∧ |
| ∀ (V : Type) [Fintype V] [Nonempty V] (G : SimpleGraph V), |
| listChromaticNumber G≤C*hadwigerNumber G ∨ |
| FractionalObstruction G := by |
| obtain ⟨K,Q,hK,hQ,hs⟩ := recursive_woven_or_obstruction |
| refine ⟨2*(Q+6*K),by omega,?_⟩ |
| intro V _ _ G |
| classical |
| by_cases hb : FractionalObstruction G |
| · exact Or.inr hb |
| left |
| have hminorfree (t : ℕ) (ht : 2≤t) (hm : hadwigerNumber G<t) : |
| listChromaticNumber G≤(Q+6*K)*t := by |
| by_contra hnot |
| have hl : (Q+6*K)*t<listChromaticNumber G := by omega |
| have hk : 1≤K*t := by nlinarith |
| obtain ⟨P,hPc,hPl⟩ := list_connectivity_extraction G hk (by nlinarith) |
| have hlist : Q*t≤listChromaticNumber (G.induce (↑P:Set V)) := by |
| change listChromaticNumber G-6*(K*t)≤ell G P at hPl |
| change Q*t≤ell G P |
| have he : (Q+6*K)*t=Q*t+6*(K*t) := by ring |
| rw [he] at hl |
| omega |
| rcases hs t (by omega) (↑P:Set V) (G.induce (↑P:Set V)) |
| (connected_induce_univ G P hPc) hlist with hw|hbad |
| swap |
| · exact hb (obstruction_induce G P hbad) |
| obtain ⟨r,hr,hrP⟩ := choose_vertices (show t≤P.card by have hh:=hPc.1; nlinarith) |
| have h0 (i : Fin 0) : V := Fin.elim0 i |
| obtain ⟨A⟩ := woven_induce G P hw r hr hrP 0 (by omega) h0 h0 |
| (fun i => Fin.elim0 i) (fun i => Fin.elim0 i) |
| have hm' : HasCliqueMinor G t := ⟨A.model.branch,A.model.connected,A.model.disjoint,A.model.adjacent⟩ |
| exact (not_lt_of_ge (cliqueMinor_le_hadwigerNumber hm')) hm |
| have hpos := one_le_hadwigerNumber G |
| have hfinal := hminorfree (hadwigerNumber G+1) (by omega) (by omega) |
| nlinarith |
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| end LinearListHadwiger.Recursion |
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| end OAI |
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