  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 $.
