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

Proved

OAI.Zarankiewicz.nonempty_polygonalExterior

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.nonempty_polygonalExterior of lean/OAI/Combinatorics/Crossing/PolygonalExterior.lean at commit adc7f12, ported to Lean 4.33.1 and Mathlib 0df444a.

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

theorem nonempty_polygonalExterior (C : DrawingPassages D) : Nonempty (PolygonalExterior C)

Definitions. The statement is written with OAI.Zarankiewicz.DrawingPassages and OAI.Zarankiewicz.FiniteIntersectionDrawing (definitions bundle OAIB3Zr000 of this split); OAI.Zarankiewicz.PolygonalExterior (definitions bundle OAIB3Zr002 of this split).

Used by. The proofs of OAI.Zarankiewicz.finite_intersection_normalization and OAI.Zarankiewicz.polygonal_no_adjacent_locally_transverse of this split import it.

Preamble

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

Formal statement

section

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} {D : FiniteIntersectionDrawing m n}

theorem nonempty_polygonalExterior (C : DrawingPassages D) : Nonempty (PolygonalExterior C) := 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/PolygonalExterior.lean (declaration OAI.Zarankiewicz.nonempty_polygonalExterior)

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