Erdos-Rogers functions for arbitrary pairs of graphs

Let f₅, ₆ (n) be the largest size of an induced F-free subgraph that every n-vertex G-free graph is guaranteed to contain. We prove that for any triangle-free graph F, \ f₅, ₊䃓 (n) = f₊䃒, ₊䃓 (n) ^1 + o (1) = n^1{2 + o (1) }. \ Along the way we give a slight improvement of a construction of Erd os-Frankl-R\"odl for the Brown-Erd os-S\'os (3r-3, 3) -problem when r is large. In contrast to our result for K₃, for any K₄-free graph F containing a cycle, we prove there exists cF > 0 such that f₅, ₊䃔 (n) > f₊䃒, ₊䃔 (n) ^1 + cF = n^1{3+cF+o (1) }. We also observe that our earlier proof for F=K₃ generalizes to f₅, ₊䃔 (n) = O (n n) for all F containing a cycle. For every graph G, we prove that there exists G >0 such that whenever F is a non-empty graph such that G is not contained in any blowup of F, then f₅, ₆ (n) = O (n^1-G). On the other hand, for graph G that is not a clique, and every >0, we exhibit a G-free graph F such that f₅, ₆ (n) = (n^1-).