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Math-2275 · mathematics · Combinatorics

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

Status Partially solved · unsolvedmath.comLLM status Open unverified

Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$, and $\mathbb C$ denote the natural numbers, integers, rationals, reals, and complex numbers, respectively. All additional parameters and hypotheses are specified in the question.
Statement recorded in the supplied source.
Let $r(C_m,K_n)$ be the least number of vertices that forces a red cycle of length $m$ or a blue clique of size $n$ in every red-blue edge-coloring of a complete graph. Is \[r(C_m,K_n)=(m-1)(n-1)+1\] for all $m\geq n$?

Formulation references: [S1], [S2].

Short English statement

Let $r(C_m,K_n)$ be the least number of vertices that forces a red cycle of length $m$ or a blue clique of size $n$ in every red-blue edge-coloring of a complete graph. Is \[r(C_m,K_n)=(m-1)(n-1)+1\] for all $m\geq n$?

Sources

Catalogue entry

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2275-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2275

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.

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

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