We say $G\to (\mathcal{C},P_n)$ if $G-E\left(F\right)$ 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 $2.066n\le \hat{R}(\mathcal{C},P_n)\le 5.25n+O\left(1\right).$ 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 $\hat{R}(\mathcal{C},P_n)\le \frac{75}{19}n+O\left(1\right)<3.947n$.

我们说 $G\to （\mathcal{C}，P_{ñ}）$ 如果 $G-Ë（F）$ 包含一个 $ñ$-顶点路径 $P_{ñ}$ 对于任何跨越森林 $F\subset G$。大小拉姆齐数$\widehat{\mathrm{[R}}（\mathcal{C}，P_{ñ}）$ 是最小的整数 $米$ 这样就存在一个图 $G$ 和 $米$ 的边缘 $G\to （\mathcal{C}，P_{ñ}）$。Dudek，Khoeini和Prałat证明了足够大$ñ$， $2。0036ñ\le \widehat{\mathrm{[R}}（\mathcal{C}，P_{ñ}）\le 31ñ$。在本说明中，我们将上限和下限都提高了$2。066ñ\le \widehat{\mathrm{[R}}（\mathcal{C}，P_{ñ}）\le 5。25ñ+Ø（\mathrm{1个}）。$我们的上限结构与Dudek，Khoeini和Prałat所考虑的完全不同。我们还有上限的计算机辅助证明$\widehat{\mathrm{[R}}（\mathcal{C}，P_{ñ}）\le \frac{75}{19}ñ+Ø（\mathrm{1个}）<3。947ñ$。