Ramsey goodness of complete multipartite graphs with one large part

For graph $G$, a connected graph $H$ of order $n$ is said to be $G$-good if $r(G,H)=(\chi(G)-1)(n-1)+s(G)$, where $\chi(G)$ is the chromatic number of $G$ and $s(G)$ is the minimum size of a color class in a $\chi(G)$-coloring of $G$. Let $K_{p+1}(\alpha;n)$ denote the complete $(p+1)$-partite graph with $p$ partite sets of size $\alpha$ and one partite set of size $n$. We determine all graphs $G$ for which $K_{p+1}(\alpha;n)$ is $G$-good for large $n$. The characterization depends on the parameter $\mathrm{snd}(\alpha)$, the smallest non-divisor of $\alpha$.