On the size Ramsey number of all cycles versus a path

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 Prałat 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 Prałat. We also have a computer assisted proof of the upper bound $\hat{R}(\mathcal{C}, P_n)\le \frac{75}{19}n +O(1) < 3.947n$.

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