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 ex(n, H) among all r-graphs 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. © 2021 Elsevier Inc.ISSN:0095-895