On the restricted size Ramsey number for a pair of cycles

TL;DR

This paper determines the minimum size of graphs with a given order that guarantee a monochromatic cycle of specified lengths in any edge coloring, focusing on the case where one cycle length is odd and large.

Contribution

It provides an exact formula for the restricted size Ramsey number for a pair of cycles when one cycle length is odd and sufficiently large.

Findings

Exact value of r^*(C_n,C_k) for large n and odd k

Extension of classical Ramsey numbers to size-restricted variants

New insights into cycle Ramsey theory

Abstract

For graphs $H_1,H_2$ by $r^*(H_1,H_2)$ we denote the minimum number of edges in a graph $G$ on $r(H_1,H_2)$ vertices such that $G\to (H_1,H_2)$. We show that for each pair of natural numbers $k,n$, $k\le n$, where $k$ is odd and $n$ is large enough, we have $$r^*(C_n,C_k)=\lceil (n+1)(2n-1)/2\rceil \,.$$