Avoiding rainbow induced subgraphs in vertex-colorings

For a fixed graph $H$ on $k$ vertices, and a graph $G$ on at least $k$ vertices, we write $G\to H$ if in any vertex-coloring of $G$ with $k$ colors, there is an induced subgraph isomorphic to $H$ whose vertices have distinct colors. In other words, if $G\to H$ then a totally multicolored induced copy of $H$ is unavoidable in any vertex-coloring of $G$ with $k$ colors. In this paper, we show that, with a few notable exceptions, for any graph $H$ on $k$ vertices and for any graph $G$ which is not isomorphic to $H$, $G\mathrm{̸}\text{\hspace{0.17em}⁣}\to H$. We explicitly describe all exceptional cases. This determines the induced vertex-anti-Ramsey number for all graphs and shows that totally multicolored induced subgraphs are, in most cases, easily avoidable.