On the set-coloring Ramsey numbers of graphs

The $\mathit{\text{set-coloring}}\mathit{\text{Ramsey}}\mathit{\text{number}}$ ${\mathrm{R}}_{r,s}(G_1,G_2,\mathrm{.}\mathrm{.}\mathrm{.},G_r)$ is the least $n\in \mathbb{N}$ such that every coloring $\chi :E\left(K_n\right)\to \left(\genfrac{}{}{0px}{}{[r]}{s}\right)$ contains a monochromatic copy of $G_i$, that is, a color $i\in [r]$ such that $i\in \chi (e)$ for every $e\in E(G_i)$. If $G_1=G_2=\cdots =G_r=G$, then we write ${\mathrm{R}}_{r,s}(G)$ for short. In 2022, Le asked to find lower and upper bounds for ${\mathrm{R}}_{s,t}(G)$ with various kinds of graphs $G$ such as stars, paths, cycles, etc. In this paper, we obtain exact values or bounds for the set-coloring Ramsey numbers of stars, paths, matchings, etc. By Lovász Local Lemma, we give a lower bound for the set-coloring Ramsey number for general graphs.