An Optimal Bound for Ramsey Goodness of Cycles
For graphs $F$ and $H$, the Ramsey number $R(F,H)$ is the minimum integer $N$ such that every $N$-vertex graph contains $F$ or its complement contains $H$. If $F$ is connected and $|F|\ge\sigma(H)$, a construction of Burr gives $R(F,H)\ge(\chi(H)-1)(|F|-1)+\sigma(H)$, where $\sigma(H)$ denotes the minimum order of a color class in a proper $\chi(H)$-coloring of $H$. Burr proved that this bound is attained for $F=C_n$ when $n$ is sufficiently large. Allen, Brightwell and Skokan conjectured that equality already holds whenever $n\geq |H| \chi(H)$, while Haslegrave, Hyde, Kim and Liu subsequently proved it whenever $n\ge C|H|\log^4\chi(H)$. Pokrovskiy and Sudakov conjectured that the optimal linear condition $n\geq C|H|$ suffices; this conjecture was also highlighted by Montgomery in his 2026 ICM survey (see Conjecture 9.2). In this paper, we resolve this conjecture by proving that there is an absolute constant $C>0$ such that $R(C_n,H)=(\chi(H)-1)(n-1)+\sigma(H)$ for every nonempty graph $H$ and every $n\ge C|H|$. This gives the first bound linear in $|H|$, and is best possible up to a constant factor. Our proof builds on the framework of Haslegrave, Hyde, Kim and Liu, and combines some new ideas in expansion and switching cycle lengths.