OpenAI research collection · 1 manuscript · Source checked 2026-10-19
The idea in plain language
In a complete network with red and blue connections, a large enough network must contain either a red cycle or a blue fully connected group. This paper claims an exact size threshold for specified cycle and group sizes, with one small exception.
Why it matters
Exact thresholds are stronger than broad estimates. The restrictions on the sizes are part of the claim.
Read the technical source claim
Proves the Erdős–Faudree–Rousseau–Schelp conjecture: R(C_m,K_n)=(m-1)(n-1)+1 for every m≥ n\ge3, except R(C_3,K_3)=6. This is the exact threshold forcing a red m-cycle or a blue n-clique in every red–blue coloring of a complete graph.
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Short excerpts below come from OpenAI's abstracts. The full abstract and original PDF preserve the paper's precise scope.
MANUSCRIPT 01
Cycle--clique Ramsey numbers
We prove that R(C_m,K_n)=(m-1)(n-1)+1 for every pair of integers m≥ n\ge3 other than (m,n)=(3,3), for which R(C_3,K_3)=6. This establishes the cycle–clique conjecture of Erdős, Faudree, Rousseau and Schelp. The proof combines expansion in a minimal counterexample with a large-clique lemma and an optimization of paths joining…
Full source abstract
We prove that R(C_m,K_n)=(m-1)(n-1)+1 for every pair of integers m≥ n\ge3 other than (m,n)=(3,3), for which R(C_3,K_3)=6. This establishes the cycle–clique conjecture of Erdős, Faudree, Rousseau and Schelp. The proof combines expansion in a minimal counterexample with a large-clique lemma and an optimization of paths joining clique vertices. These arguments reduce the remaining cases to 3{,}099 finite parameter-pattern instances, which are excluded by two exact implementations of proved inference rules. Complete programs and deduction traces accompany the paper.
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New mathematical reasoning could help AI tackle harder scientific and computational problems. How reliably these skills transfer beyond mathematics will determine how much they contribute to general intelligence.