Cycle-Complete Ramsey Numbers. (10th July 2019) - Keevash, Peter; Long, Eoin; Skokan, Jozef
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, Erd̋s, Faudree, Rousseau, and Schelp conjectured that $r(C_{\ell }, K_n) = (\ell -1)(n-1)+1$ for $\ell ≥ n≥ 3$ provided $(\ell, n) ≠ (3, 3)$ . We prove that, for some absolute constant $C\ge 1$, we have $r(C_{\ell }, K_n) = (\ell -1)(n-1)+1$ provided $\ell ≥ C\frac{\log n}{\log \log n}$ . Up to the value of $C$ this is tight since we also show that, for any $ε>0$ and $n> n_0(ε )$, we have $r(C_{\ell }, K_n) \gg (\ell -1)(n-1)+1$ for all $3 ≤ \ell ≤ (1-ε )\frac{\log n}{\log \log n}$ . This proves the conjecture of Erd̋s, 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 Erd̋s, Faudree, Rousseau, and Schelp.