Maximum $\mathcal H$-free subgraphs

Abstract

Given a family of hypergraphs H, let f (m, H) denote the largest size of an H-free subgraph that one is guaranteed to find in every hypergraph with m edges. This function was first introduced by Erdős and Komlós in 1969 in the context of union-free families, and various other special cases have been extensively studied since then. In an attempt to develop a general theory for these questions, we consider the following basic issue: which sequences of hypergraph families {Hm} have bounded f (m, Hm) as m → ∞? A variety of bounds for f (m, Hm) are obtained which answer this question in some cases. Obtaining a complete description of sequences {Hm} for which f (m, Hm) is bounded seems hopeless.