1 · The essentials
Imagine a collection of points with every pair joined by a line, and color each line either red or blue. We ask how many points guarantee either a red loop through exactly distinct points or a group of points whose connecting lines are all blue. The result gives the exact answer whenever , with one exception. That answer is . The exception is when both requested groups have size three: then the answer is six. “Exact” means that every coloring at the stated size contains one of the requested arrangements, while some coloring with one fewer point contains neither. The main result has a Lean formalization.
A picture to help (an imperfect image, not to be taken literally)
As an imperfect image, imagine people at a gathering, with each pair labeled red or blue. You seek either a round-table ordering of people with red labels between consecutive neighbors, including the last and first, or people with blue labels between every pair. The image is only about pairwise labels, not about how people actually behave.
Why it matters
The result settles the cycle–clique conjecture attributed in the source to Erdős, Faudree, Rousseau and Schelp. It supplies an exact threshold across the whole stated range, rather than only an estimate. The manuscript also connects structural reasoning about graphs with a finite check accompanied by programs and deduction traces.
2 · Going further
The Ramsey number is the smallest number of vertices such that every red–blue coloring of the edges of a complete graph contains a red cycle or a blue clique . The main result is for integers , except that . A cycle uses distinct vertices joined in a closed sequence; a clique requires an edge between every pair of its vertices, and here all those edges must be blue.
The manuscript describes the argument as combining expansion in a minimal counterexample with a large-clique lemma and an optimization of paths joining clique vertices. In outline, it studies the structure a smallest failure would have to possess, then uses clique and path constraints to narrow the possibilities. The abstract does not give the precise expansion conditions or the statement of the large-clique lemma.
These arguments reduce the remaining cases to finite parameter-pattern instances. The manuscript reports that two exact implementations of its inference rules exclude those instances, and that complete programs and deduction traces accompany the paper. The main result has a Lean formalization, but the supplied material does not specify whether that formalization covers every auxiliary claim or computational artifact. Only the main manuscript is supplied here, so no relationship to companion manuscripts can be described.
Useful notions
- Graphs, vertices, and edges
- Complete graphs and edge colorings
- Cycles and cliques
- Exact thresholds
- Minimal-counterexample reasoning
What was known before / what is new
Before
The source identifies the stated formula as the cycle–clique conjecture of Erdős, Faudree, Rousseau and Schelp. It gives no account of earlier partial results or the conjecture’s chronology.
New
The main result establishes the conjectured formula throughout the stated range, with the specified exception at . The manuscript describes a combination of expansion, a large-clique lemma, and path optimization that reduces the remaining work to a finite collection of instances. It supplies complete programs and deduction traces for the reported exclusions.
Limits and assumptions
The formula applies to integers and has the explicit exception . It concerns red–blue edge colorings of complete graphs; no claim about other parameter ranges or coloring settings is supplied. The main result has a Lean formalization, but its detailed scope is not stated in the supplied material, so coverage of every lemma, implementation, and deduction trace should not be assumed. The source lists no remaining open problems.
Mini glossary
- Complete graph —
- A graph in which every pair of distinct vertices is joined by an edge.
- Cycle —
- A closed sequence of edges through distinct vertices, returning from the last vertex to the first.
- Clique —
- A set of vertices with an edge between every pair, with a blue clique requiring all those edges to be blue.
- Ramsey number —
- Here, the smallest number of vertices forcing a red cycle of the requested length or a blue clique of the requested size in every red–blue coloring.
- Minimal counterexample —
- A smallest object that would violate the claimed result, studied to constrain what a failure could look like.
Keywords : Ramsey numbers · Cycles · Cliques
AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.
3 · Technical details and sources
Claim made in the manuscript (source text)
Proves the Erdős–Faudree–Rousseau–Schelp conjecture: for every , except . This is the exact threshold forcing a red m-cycle or a blue n-clique in every red–blue coloring of a complete graph.
Verified by Lean: the Lean software checked the part of the reasoning described below, sometimes only part of the manuscript. This does not mean independent mathematicians have validated the result.
Manuscripts in this family
Cycle--clique Ramsey numbers
25/09/2026We prove that for every pair of integers other than , for which . 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 finite parameter-pattern instances, which are excluded by two exact implementations of proved inference rules. Complete programs and deduction traces accompany the paper.