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Math-1142-A1 · revision r1 · on Math-1142

OpenAI claim summary for Math-1142

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 graph crossing numbers: For every complete graph K_n the claimed crossing number is exactly the Hill four-floor-factor expression, proving both the exact-value records and the matching lower-bound formulation.

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. The Crossing Number of the Complete Graph
  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

The Crossing Number of the Complete Graph

OpenAI's claim

Complete graph crossing numbers: The paper claims that the ordinary crossing number of $K_n$ equals one quarter of the product of the four consecutive half-integer floor factors. The statement holds at every positive integer order.

Mathematical statement of the claimed result

Mathematical claim 1.
For every integer $n\ge3$, the manuscript asserts \[\operatorname{cr}(K_n)=\frac14\left\lfloor\frac n2\right\rfloor\left\lfloor\frac{n-1}2\right\rfloor\left\lfloor\frac{n-2}2\right\rfloor\left\lfloor\frac{n-3}2\right\rfloor.\] Here $\operatorname{cr}$ minimizes the number of proper edge-crossing points over ordinary plane drawings, allowing curved edges.

Outline of the argument

Complete graph crossing numbers: The drawing is reduced to a controlled simple-arc form, then signed cycle intersections and endpoint orders are encoded algebraically. An even-node order detector supplies the rank obstruction giving the lower bound. A two-page drawing attains the proposed value, with the normalization argument ensuring that unrestricted continuous-arc drawings are covered.

Formal verification

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

Complete graph 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