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d5b7741 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 | 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
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