Cycle-complete Ramsey numbers

TL;DR

This paper determines the exact Ramsey number for cycles versus cliques for large cycle lengths, confirming a longstanding conjecture for a broad range of cycle sizes and advancing understanding in combinatorial graph theory.

Contribution

It proves the conjecture of Erdős, Faudree, Rousseau, and Schelp for large cycle lengths, establishing the exact value of the cycle-complete Ramsey number in that regime.

Findings

Proves $r(C_{ ext{ell}},K_n) = ( ext{ell}-1)(n-1)+1$ for $ ext{ell} ext{ large}$

Shows the conjecture holds for $ ext{ell} o rac{ ext{log} n}{ ext{log} ext{log} n}$

Provides bounds indicating the conjecture's tightness for smaller $ ext{ell}$

Abstract

The Ramsey number $r(C_{\ell},K_n)$ is the smallest natural number $N$ such that every red/blue edge-colouring of a clique of order $N$ contains a red cycle of length $\ell$ or a blue clique of order $n$. In 1978, Erd\H{o}s, Faudree, Rousseau and Schelp conjectured that $r(C_{\ell},K_n) = (\ell-1)(n-1)+1$ for $\ell \geq n\geq 3$ provided $(\ell,n) \neq (3,3)$. We prove that, for some absolute constant $C\ge 1$, we have $r(C_{\ell},K_n) = (\ell-1)(n-1)+1$ provided $\ell \geq C\frac {\log n}{\log \log n}$. Up to the value of $C$ this is tight since we also show that, for any $\varepsilon >0$ and $n> n_0(\varepsilon )$, we have $r(C_{\ell }, K_n) \gg (\ell -1)(n-1)+1$ for all $3 \leq \ell \leq (1-\varepsilon )\frac {\log n}{\log \log n}$. This proves the conjecture of Erd\H{o}s, Faudree, Rousseau and Schelp for large $\ell $, a stronger form of the conjecture due to Nikiforov, and answers (up to multiplicative constants) two further questions of Erd\H{o}s, Faudree, Rousseau and Schelp.