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arXiv · 1905.01709

Maximum $\mathcal H$-free subgraphs

Dhruv Mubayi·Sayan Mukherjee
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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.

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BibTeXRIS

Dhruv Mubayi, Sayan Mukherjee. 2019-05-05. Maximum $\mathcal H$-free subgraphs. https://doi.org/10.4310/joc.2021.v12.n2.a1

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