Given r -uniform hypergraphs G and F and an integer n , let f F , G ( 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 f F , G ( 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 f F , G ( 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. (Fri,) studied this question.