The uniform Turán density $\pi_u(F)$ of a hypergraph $F$, introduced by Erdős and Sós, is the smallest value of $d$ such that any hypergraph $H$ where all linear-sized subsets of vertices of $H$ have density greater than $d$ contains $F$ as a subgraph. Over the past few years the value of $\pi_u(F)$ was determined for several classes of 3-graphs, but no nonzero value of $\pi_u(F)$ has been found for $r$-graphs with $r>3$. In this talk we show the existence of $r$-graphs $F$ with $\pi_u(F)={r \choose 2}^{-{r \choose 2}}$, which we conjecture is minimum possible. Joint work with Frederik Garbe, Daniel Il’kovic, Dan Král’ and Filip Kučerák.
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