Ramsey numbers of paths and graphs of the same order

For graphs $F_n$ and $G_n$ of order $n$, if $R(F_n, G_n)=(χ(G_n)-1)(n-1)+σ(G_n)$, then $F_n$ is said to be $G_n$-good, where $σ(G_n)$ is the minimum size of a color class among all proper vertex-colorings of $G_n$ with $χ(G_n)$ colors. Given $Δ(G_n)\le Δ$, it is shown that $P_n$ is asymptotically $G_n$-good if $α(G_n)\le\frac{n}{4}$.

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