| import OAI.Combinatorics.Crossing.AxisConcurrency | |
| namespace OAI | |
| noncomputable section | |
| namespace Zarankiewicz | |
| variable {m n : ℕ} | |
| def AxisIndependentTriple (e f g : BipartiteEdge m n) : Prop := | |
| e.1 ≠ f.1 ∧ e.1 ≠ g.1 ∧ f.1 ≠ g.1 ∧ e.2 ≠ f.2 ∧ e.2 ≠ g.2 ∧ f.2 ≠ g.2 | |
| theorem exists_axis_generic_positions (lo hi : BipartiteVertex m n → ℝ) | |
| (hinterval : ∀ j, lo j < hi j) : | |
| ∃ x : BipartiteVertex m n → ℝ, (∀ j, x j ∈ Set.Ioo (lo j) (hi j)) ∧ | |
| ∀ e f g : BipartiteEdge m n, AxisIndependentTriple e f g → | |
| axisConcurrency (x (Sum.inl e.1)) (x (Sum.inr e.2)) (x (Sum.inl f.1)) (x (Sum.inr f.2)) | |
| (x (Sum.inl g.1)) (x (Sum.inr g.2)) ≠ 0 := by | |
| classical | |
| let P : (BipartiteEdge m n × BipartiteEdge m n × BipartiteEdge m n) → MvPolynomial (BipartiteVertex m n) ℝ := | |
| fun t => if AxisIndependentTriple t.1 t.2.1 t.2.2 then axisConcurrencyPolynomial t.1 t.2.1 t.2.2 else 1 | |
| have hP : ∀ t, P t ≠ 0 := by | |
| intro t | |
| dsimp only [P] | |
| split_ifs with h | |
| · exact axisConcurrencyPolynomial_ne_zero _ _ _ h.1 (Ne.symm h.2.2.1) | |
| (Ne.symm h.2.2.2.1) (Ne.symm h.2.2.2.2.1) | |
| · exact one_ne_zero | |
| obtain ⟨x,hx,havoid⟩ := exists_box_avoiding_polynomials P hP lo hi hinterval | |
| refine ⟨x,hx,?_⟩ | |
| intro e f g htriple | |
| have hh := havoid (e,f,g) | |
| simpa only [P,ite_eq_left htriple,axisConcurrencyPolynomial_eval] using hh | |
| end Zarankiewicz | |
| end | |
| end OAI | |