On the size Ramsey number of all cycles versus a path

TL;DR

This paper improves bounds on the size Ramsey number for all cycles versus a path, narrowing the range and introducing a new construction and computer-assisted proof.

Contribution

It provides tighter bounds on the size Ramsey number for cycles versus a path, with a novel construction and computational verification.

Findings

Lower bound improved to 2.066n

Upper bound improved to approximately 3.947n

New construction differs from previous methods

Abstract

We say $G\to (\mathcal{C}, P_n)$ if $G-E(F)$ contains an $n$-vertex path $P_n$ for any spanning forest $F\subset G$. The size Ramsey number $\hat{R}(\mathcal{C}, P_n)$ is the smallest integer $m$ such that there exists a graph $G$ with $m$ edges for which $G\to (\mathcal{C}, P_n)$. Dudek, Khoeini and Pra{\l}at proved that for sufficiently large $n$, $2.0036n \le \hat{R}(\mathcal{C}, P_n)\le 31n$. In this note, we improve both the lower and upper bounds to $2.066n\le \hat{R}(\mathcal{C}, P_n)\le 5.25n+O(1).$ Our construction for the upper bound is completely different than the one considered by Dudek, Khoeini and Pra{\l}at. We also have a computer assisted proof of the upper bound $\hat{R}(\mathcal{C}, P_n)\le \frac{75}{19}n +O(1) < 3.947n $.