On the maximum F-free induced subgraphs in Kₜ-free graphs

For graphs F and H, let f₅, ₇ (n) be the minimum possible size of a maximum F-free induced subgraph in an n-vertex H-free graph. This notion generalizes the Ramsey function and the Erdos--Rogers function. Establishing a container lemma for the F-free subgraphs, we give a general upper bound on f₅, ₇ (n), assuming the existence of certain locally dense H-free graphs. In particular, we prove that for every graph F with ex (m, F) = O (m^1+), where 0, 1/2), we have \[ f₅, ₊䃓 (n) = O (n^1{2-} (n) ^3{2- }) and f₅, ₊䃔 (n) = O (n^1{3-2} (n) ^6{3-2}). \ For the cases where F is a complete multipartite graph, letting s = ₈=₁ʳ sᵢ, we prove that \ f₊_ₒ䃑, , ₒ㶂, Kₑ+₂ (n) = O (n^2s -3{4s -5} (n) ^3). \ We also make an observation which improves the bounds of ex (G (n, p), C₄) by a polylogarithmic factor.