# Maximum $\mathcal H$-free subgraphs

## TL;DR

This paper investigates the maximum size of hypergraphs free of certain substructures, establishing bounds and exploring conditions under which these bounds remain finite as the hypergraph size grows.

## Contribution

The paper develops bounds for the function $f(m, \\mathcal{H}_m)$ and explores the conditions for boundedness across sequences of hypergraph families.

## Key findings

- Derived bounds for $f(m, \\mathcal{H}_m)$ in various cases
- Identified conditions for boundedness of $f(m, \\mathcal{H}_m)$ as $m$ increases
- Showed that a complete characterization of bounded sequences is unlikely

## Abstract

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.