Given r-uniform hypergraphs G and F and an integer n, let f₅, ₆ (n) be the maximum m such that every n-vertex G-free r-graph has an F-free induced subgraph on m vertices. We show that f₅, ₆ (n) is polynomial in n when G is a subgraph of an iterated blowup of F. As a partial converse, we show that if G is not a subgraph of an F-iterated blowup and is 2-tightly connected, then f₅, ₆ (n) is at most polylogarithmic in n. Our bounds generalize previous results of Dudek and Mubayi for the case when F and G are complete.
He et al. (Thu,) studied this question.