Uniform Turán density beyond 3-graphs

Abstract

In the 1980s, Erdős and Sós first introduced an extremal problem on hypergraphs with density constraints. Given an $r$-uniform hypergraph $F$ (or $r$-graph for short), its uniform Turán density $π_u(F)$ is the smallest value of $d$ in which every hypergraph $H$ in which every linear-sized subhypergraph of $H$ has edge density at least $d$ contains $F$ as a subgraph. The first non-zero value of $π_u(F)$ was not found until 30 years later. Progress in studying the set of values of the uniform Turán density of $r$-graphs has been uneven in terms of $r$: to this day there are infinitely many non-zero values known for $r=3$, a single non-zero value known for $r=4$ and none for $r\geq 5$. In this paper we obtain the first explicit values of $π_u$ for all uniformities, by proving that for every $r\geq 3$ there exist $r$-graphs $F$ with $π_u(F)=1/4$ and with $π_u(F)=\binom{r}{2}^{-\binom{r}{2}}$.