Unavoidable hypergraphs

The following very natural problem was raised by Chung and Erdős in the early 80's and has since been repeated a number of times. What is the minimum of the Turán 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 Erdős 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.