Extremal problems in uniformly dense hypergraphs and digraphs

The uniform Tur\'an density $\pi_{u}(F)$ of a $3$-uniform hypergraph (or $3$-graph) $F$ is the supremum of all $d$ such that there exist infinitely many $F$-free $3$-graphs $H$ in which every induced subhypergraph on a linearly sized vertex set has edge density at least $d$. Determining $\pi_{u}(F)$ for a given $3$-graph $F$ was proposed by Erd\H{o}s and S\'os in the 1980s, yet only a few cases are known. In particular, it remains open whether $1/2$ can occur as a value of $\pi_{u}$. In this paper, we establish a novel connection between Tur\'an-type extremal problems for digraphs and uniform Tur\'an densities of $3$-graphs. Using digraph extremal results, we give the first verifiable conditions for $3$-graphs $F$ with $\pi_{u}(F) = (r-1)/r$ and $\pi_{u}(F) = (r-1)^2/r^2$ for all $r \ge 2$, and identify the corresponding $3$-graphs. In particular, these $3$-graph classes contain some specific $3$-graphs, such as $K^{(3)-}_4$. We also present a sufficient condition ensuring $\pi_{u}(F)=4/27$ and construct $3$-graphs satisfying it; in particular, our examples are different from the tight $3$-uniform cycles whose uniform Tur\'an density $4/27$ was determined in [{Trans. Amer. Math. Soc. 376 (2023), 4765-4809}]. Finally, we give a short proof of the existence of $3$-graphs $F$ with $\pi_{u}(F)=1/27$, originally established by Garbe, Kr\'al' and Lamaison [{Israel J. Math. 259 (2024), 701-726}] via the hypergraph regularity method.