# Maximum H-free subgraphs

Published in *Journal of Combinatorics*, 2021

Recommended citation: Mubayi, Dhruv and Mukherjee, Sayan. "Maximum H-free subgraphs." *Journal of Combinatorics* Volume 12 (2) Pages 185-214 (2021). __https://doi.org/10.4310/JOC.2021.v12.n2.a1__

Given a family of hypergraphs $\mathcal H$, let $f(m,\mathcal H)$ denote the largest size of an $\mathcal H$-free subgraph that one is guaranteed to find in every hypergraph with $m$ edges. This function was first introduced by Erd\H{o}s and Koml'{o}s 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 ${\mathcal H_m}$ have bounded $f(m,\mathcal H_m)$ as $m\to\infty$? A variety of bounds for $f(m,\mathcal H_m)$ are obtained which answer this question in some cases. Obtaining a complete description of sequences ${\mathcal H_m}$ for which $f(m,\mathcal H_m)$ is bounded seems hopeless.