OAI.Zarankiewicz.polygonal_no_adjacent_locally_transverse: intermediate lemma from OpenAI, The crossing number of complete bipartite graphs

Proved

OAI.Zarankiewicz.polygonal_no_adjacent_locally_transverse

by dbenbenn · Oct 25, 2026 · Mathlib 0df444a (Lean v4.33.1)

openai-math

An intermediate lemma of the split of OpenAI's proof of OAI.Zarankiewicz.mainTarget_proof, carved from OpenAI's openai/math development for this platform. The statement and its proof are OpenAI's (OpenAI, The crossing number of complete bipartite graphs, OpenAI Math Release, September 23, 2026, result family 165, github.com/openai/math, Apache License 2.0); in Lean it is the declaration OAI.Zarankiewicz.polygonal_no_adjacent_locally_transverse of lean/OAI/Combinatorics/Crossing/LocallyTransverseNormalization.lean at commit adc7f12, ported to Lean 4.33.1 and Mathlib 0df444a.

The statement, as published (namespace OAI.Zarankiewicz):

theorem polygonal_no_adjacent_locally_transverse (D : AdmissibleDrawing m n) :
    ∃ A : AdmissibleDrawing m n, crossingCount A ≤ crossingCount D ∧
      (∀ e, HasFiniteAffineCover (A.edge e)) ∧ NoAdjacentCrossings A ∧
      LocallyStraightCrossings A ∧ LocallyTransverseCrossings A

Definitions. The statement is written with OAI.Zarankiewicz.NoAdjacentCrossings (definitions bundle OAIB3Zr001 of this split); OAI.Zarankiewicz.HasFiniteAffineCover, OAI.Zarankiewicz.LocallyStraightCrossings and OAI.Zarankiewicz.LocallyTransverseCrossings (definitions bundle OAIB3Zr002 of this split); OAI.Zarankiewicz.AdmissibleDrawing, OAI.Zarankiewicz.BipartiteEdge, OAI.Zarankiewicz.crossingCount and OAI.Zarankiewicz.ContinuousDrawing (the published bundle Def_BipartiteCrossing).

Used by. The proof of OAI.Zarankiewicz.mainTarget_proof of this split imports it.

Preamble

import Mathlib
import Lean.Elab.Tactic.Omega
import Definitions.Def_BipartiteCrossing
import Definitions.Def_OAIB3Zr002

Formal statement

section

attribute [-instance] _root_.OAI.Zarankiewicz.incidentEdgesFintype
set_option linter.all false
set_option linter.unusedVariables false
set_option linter.unusedTactic false
set_option linter.unreachableTactic false

namespace OAI

noncomputable section
namespace Zarankiewicz
variable {m n : ℕ}

theorem polygonal_no_adjacent_locally_transverse (D : AdmissibleDrawing m n) :
    ∃ A : AdmissibleDrawing m n, crossingCount A ≤ crossingCount D ∧
      (∀ e, HasFiniteAffineCover (A.edge e)) ∧ NoAdjacentCrossings A ∧
      LocallyStraightCrossings A ∧ LocallyTransverseCrossings A := by
  sorry

end Zarankiewicz

end

end OAI
end

Source

OpenAI, The crossing number of complete bipartite graphs, OpenAI Math Release, September 23, 2026, https://github.com/openai/math (result family 165; paper: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-crossing-number-of-complete-bipartite-graphs-September-23-2026/paper.pdf; Apache License 2.0); Lean: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Combinatorics/Crossing/LocallyTransverseNormalization.lean (declaration OAI.Zarankiewicz.polygonal_no_adjacent_locally_transverse)

View graph