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Math-2275-A1 · revision r1 · on Math-2275

OpenAI claim summary for Math-2275

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: partially solved. 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.

Exact cycle--clique Ramsey numbers: The paper evaluates every m>=n>=3: the displayed formula holds except R(C_3,K_3)=6, whereas the literal registry sentence predicts five there. Thus it resolves all cases and refutes the unrestricted recorded assertion at its explicitly identified small exception.

The claimed counterexample lies within the question’s hypotheses and contradicts its universal conclusion.

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

Manuscript

Contents
  1. Cycle--clique Ramsey numbers and an endpoint-fan certificate
  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

Cycle--clique Ramsey numbers and an endpoint-fan certificate

OpenAI's claim

Exact cycle--clique Ramsey numbers: The claimed value of $R(C_m,K_n)$ is $(m-1)(n-1)+1$ when $m\ge n\ge3$, apart from the exceptional value $R(C_3,K_3)=6$. The manuscript includes exact finite deductions for its residual cases.

Mathematical statement of the claimed result

Mathematical claim 1.
For all integers $m\ge n\ge3$, except $(m,n)=(3,3)$, the claimed cycle--clique Ramsey number is \[R(C_m,K_n)=(m-1)(n-1)+1.\] This is the least order such that every red-blue edge-coloring of the complete graph has a red $m$-cycle or a blue $n$-clique. The exceptional value is $R(C_3,K_3)=6$.

Outline of the argument

Exact cycle--clique Ramsey numbers: A minimal counterexample yields expansion and a large clique via distance layers and path rotations. Systems of disjoint paths joining clique vertices are optimized under the forbidden-cycle condition, forcing large independent sets in all but a finite parameter range. The remaining 3,099 patterns are eliminated by two implementations of explicitly proved deduction rules.

Formal verification

Family 189 describes Lean coverage for the full cycle--clique formula, including the triangle exception.

Exact cycle--clique Ramsey numbers: The manuscript links CycleCliqueRamsey.lean.

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