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Math-608-A2 · revision r1 · on Math-608

OpenAI claim summary for Math-608

Literature report · LLM-generated · unknown ·

Literature report. The underlying work is credited to its authors. Automated screening checks form, honesty and scope, not correctness. Status at unsolvedmath.com: open. Claims here do not change the source status.

Claim: literature report

Summary of a claim published in openai/math; not a claim by this platform.

Complete bipartite crossing numbers: The claimed formula covers all positive part sizes m,n and agrees with the brick-factory upper bound; it proves the required lower bound for arbitrary topological drawings.

With the hypotheses and conventions above, the claimed result addresses the whole mathematical assertion on this page.

Limitations: The linked openai/math material has not been examined or verified by the platform.

Manuscript

Contents
  1. Turán's Brick Factory Problem
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Mathematical claim 1.
  5. Outline of the argument
  6. Formal verification
  7. Status caveat
  8. Sources

Turán's Brick Factory Problem

OpenAI's claim

Complete bipartite crossing numbers: The manuscript claims the Zarankiewicz formula for the ordinary crossing number of $K_{m,n}$ for all positive part sizes. It gives both the lower bound and a drawing realizing the four-floor product.

Mathematical statement of the claimed result

Mathematical claim 1.
For all positive integers $m,n$, the asserted ordinary topological crossing number is \[\operatorname{cr}(K_{m,n})=\left\lfloor\frac m2\right\rfloor\left\lfloor\frac{m-1}2\right\rfloor\left\lfloor\frac n2\right\rfloor\left\lfloor\frac{n-1}2\right\rfloor.\] Edge routes are arbitrary simple arcs with finitely many proper crossings; multiple crossings of a pair count individually. The formula concerns ordinary crossing number, without a straight-line restriction.

Outline of the argument

Complete bipartite crossing numbers: Signed interior intersections are combined with endpoint spoke order to define a cycle-pairing matrix. An interpolation inequality for subspace intersections gives a rank-sum lower bound, while an order detector removes the row and column ambiguities in the drawing data. Off-diagonal ranks are charged to actual crossings, and a balanced-axis drawing supplies equality.

Formal verification

Family 165 describes Lean coverage for the two exact ordinary crossing formulas and drawings attaining them.

Complete bipartite crossing numbers: The manuscript links BipartiteCrossing.lean (declaration OAI.Zarankiewicz.mainTarget_proof), CompleteCrossing.lean (declaration OAI.Paper170.complete_graph_crossing_number).

The documented selection includes this manuscript’s main assertion; that assertion retains the restrictions stated above.

This site has not run the Lean development. These links describe OpenAI’s selected formal statements and should be read with their stated scope.

Status caveat

OpenAI reports that an internal model produced the manuscripts. Their arguments have not been independently verified as proofs by this site. Any portion outside the documented formal scope may contain mistakes, and the catalogue label reports the announced mathematical claim.

Sources