A note on the size Ramsey number of powers of paths

Let $r\geq3$ be an integer such that $r-2$ is a prime power and let $H$ be a connected graph on $n$ vertices with average degree at least $d$ and $\alpha(H)\leq\beta n$, where $0<\beta<1$ is a constant. We prove that the size Ramsey number $$\hat{R}({H};r) > \frac{{nd}}{2}{(r - 2)^2} - C\sqrt n$$ for all sufficiently large $n$, where $C$ is a constant depending only on $r$ and $d$. In particular, for integers $k\ge1$, and $r\ge3$ such that $r-2$ is a prime power, we have that there exists a constant $C$ depending only on $r$ and $d$ such that $\hat{R}(P_{n}^{k}; r)> kn{(r - 2)^2}-C\sqrt n -\frac{{({k^2} + k)}}{2}{(r - 2)^2}$ for all sufficiently large $n$, where $P_{n}^{k}$ is the $kth$ power of $P_n$. We also prove that $\hat{R}(P_n,P_n,P_n)<764.1n$ for sufficiently large $n$. This result improves some results of Dudek and Pra{\l}at (\emph{SIAM J. Discrete Math.}, 31 (2017), 2079--2092 and \emph{Electron. J. Combin.}, 25 (2018), no.3, # P3.35).