Result 189, Combinatorics

Cycle–clique Ramsey numbers

Proves the Erdős–Faudree–Rousseau–Schelp conjecture: R(Cm,Kn)=(m−1)(n−1)+1R(C_m,K_n)=(m-1)(n-1)+1 for every m≥n≥3m\ge n\ge3, except R(C3,K3)=6R(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.

Lean formalization Proof

The bigger picture

Why it matters

Coloring every connection in a network red or blue cannot indefinitely avoid both red loops and blue fully connected groups. An unreviewed manuscript claims an exact size threshold for when one of these patterns becomes unavoidable.

What changes?

The manuscript reports the exact threshold for complete graphs, in which every pair of vertices is connected. Every red-blue edge coloring must contain either a red cycle, a closed loop through m distinct vertices, or a blue clique, n vertices with all their connecting edges blue. For integers m at least n at least 3, the threshold is one plus (m minus one) times (n minus one), except when both equal 3: then it is 6.

What does that help mathematicians do?

Outside the exceptional pair, the formula matches a simple way to avoid both patterns: split the vertices into n minus one groups of m minus one vertices, color edges within groups red and edges between groups blue. Red cycles cannot be long enough, and a blue clique can use only one vertex per group. The claimed result says that adding just one vertex makes avoidance impossible, regardless of the coloring.

Are there practical applications?

Its immediate value is foundational: an exact boundary for forced structure in colored graphs. The reported proof reduces the remaining cases to 3,099 finite parameter-pattern instances, ruled out using two exact implementations of its inference rules. Accompanying programs and deduction traces give researchers concrete material for scrutinizing the computational part of the argument.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

Cycle--clique Ramsey numbers

September 25, 2026 41 pages

We prove that R(Cm,Kn)=(m−1)(n−1)+1R(C_m,K_n)=(m-1)(n-1)+1 for every pair of integers m≥n≥3m\ge n\ge3 other than (m,n)=(3,3)(m,n)=(3,3), for which R(C3,K3)=6R(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 3,0993{,}099 finite parameter-pattern instances, which are excluded by two exact implementations of proved inference rules. Complete programs and deduction traces accompany the paper.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Cycle-clique-Ramsey-numbers-September-25-2026,
  author = {{OpenAI}},
  title = {{Cycle--clique Ramsey numbers}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Cycle-clique-Ramsey-numbers-September-25-2026/Cycle-clique-Ramsey-numbers-September-25-2026.pdf}{OAI:Cycle-clique-Ramsey-numbers-September-25-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/189.md.

Cycle–clique Ramsey numbers

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

The cycle–clique Ramsey conjecture predicts the exact number of vertices forcing either a cycle CmC_m or a clique KnK_n in complementary colors. The formalization proves R(Cm,Kn)=(m−1)(n−1)+1R(C_m,K_n)=(m-1)(n-1)+1 for all integers m≥n≥3m\ge n\ge3 except (m,n)=(3,3)(m,n)=(3,3), where it proves R(C3,K3)=6R(C_3,K_3)=6. This is the complete parameter range of the paper's main theorem.

Comparator links

Result Comparator statement
Exact cycle–clique Ramsey numbers CycleCliqueRamsey.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 8 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.