On size Ramsey numbers for a pair of cycles

TL;DR

This paper establishes an upper bound on the size Ramsey number for pairs of cycles, specifically showing it is linearly bounded by the cycle length, and explores related restricted size Ramsey numbers.

Contribution

The paper proves a linear upper bound on the size Ramsey number for pairs of cycles and investigates the restricted size Ramsey number for these pairs.

Findings

Size Ramsey number for pairs of cycles is linearly bounded by cycle length.

Existence of an absolute constant A bounding the size Ramsey number.

Analysis of restricted size Ramsey numbers for cycle pairs.

Abstract

We show that there exists an absolute constant $A$ such that the size Ramsey number of a pair of cycles $(C_n$, $C_{2d})$, where $4\le 2d\le n$, is bounded from above by $An$. We also study the restricted size Ramsey number for such a pair.