Ramsey numbers of paths and graphs of the same order

For graphs $F_n$ and $G_n$ of order $n$, if $R(F_n, G_n)=(\chi(G_n)-1)(n-1)+\sigma(G_n)$,then$F_n$issaidtobe$G_n$-good, where $\sigma (G_n)$ is the minimum size of a color class among all proper vertex-colorings of $G_n$ with $\chi (G_n)$ colors. Given $\Delta(G_n)\le \Delta$,itisshownthat$P_n$isasymptotically$G_n$-goodif$\alpha(G_n)\le\frac{n}{4}$.