Rainbow induced subgraphs in proper vertex colorings

For a given graph $H$ we define $\rho (H)$ to be the minimum order of a graph $G$ such that every proper vertex coloring of $G$ contains a rainbow induced subgraph isomorphic to $H$. We give upper and lower bounds for $\rho (H)$, compute the exact value for some classes of graphs, and consider an interesting combinatorial problem connected with computation of $\rho (H)$ for paths. This research is motivated by some ideas in on-line graph coloring algorithms.