On the Maximum F-Free Induced Subgraphs in K_t-Free Graphs

For graphs (Formula presented.) and (Formula presented.), let (Formula presented.) be the minimum possible size of a maximum (Formula presented.) -free induced subgraph in an (Formula presented.) -vertex (Formula presented.) -free graph. This notion generalizes the Ramsey function and the Erdős–Rogers function. Establishing a container lemma for the (Formula presented.) -free subgraphs, we give a general upper bound on (Formula presented.), assuming the existence of certain locally dense (Formula presented.) -free graphs. In particular, we prove that for every graph (Formula presented.) with (Formula presented.), where (Formula presented.), we have (Formula presented.) For the cases where (Formula presented.) is a complete multipartite graph, letting (Formula presented.), we prove that (Formula presented.) We also make an observation which improves the bounds of (Formula presented.) by a polylogarithmic factor.