Unavoidable hypergraphs

Abstract

The following very natural problem was raised by Chung and Erd\H{o}s in the early 80's and has since been repeated a number of times. What is the minimum of the Tur\'an number $\text{ex}(n,\mathcal{H})$ among all $r$-graphs $\mathcal{H}$ with a fixed number of edges? Their actual focus was on an equivalent and perhaps even more natural question which asks what is the largest size of an $r$-graph that can not be avoided in any $r$-graph on $n$ vertices and $e$ edges? In the original paper they resolve this question asymptotically for graphs, for most of the range of $e$. In a follow-up work Chung and Erd\H{o}s resolve the $3$-uniform case and raise the $4$-uniform case as the natural next step. In this paper we make first progress on this problem in over 40 years by asymptotically resolving the $4$-uniform case which gives us some indication on how the answer should behave in general.