Size-Ramsey numbers of cycles versus a path

The size-Ramsey number $\hat{R}(\mathcal{F},H)$ of a family of graphs $\mathcal{F}$ and a graph $H$ is the smallest integer $m$ such that there exists a graph $G$ on $m$ edges with the property that any colouring of the edges of $G$ with two colours, say, red and blue, yields a red copy of a graph from $\mathcal{F}$ or a blue copy of $H$. In this paper we first focus on $\mathcal{F} = \mathcal{C}_{\le cn}$, where $\mathcal{C}_{\le cn}$ is the family of cycles of length at most $cn$, and $H = P_n$. In particular, we show that $2.00365 n \le \hat{R}(\mathcal{C}_{\le n},P_n) \le 31n$. Using similar techniques, we also managed to analyze $\hat{R}(C_n,P_n)$, which was investigated before but only using the regularity method.