Extremal problems for uniformly dense hypergraphs

Extremal problems for hypergraphs concern the maximum
density of large hypergraphs $H$ that do not contain
a copy of a given hypergraph $F$. Estimating the so-called
Tur’an-densities is a central problem in combinatorics.
However, despite a lot of effort precise estimates
are only known for very few hypergraphs~$F$.
We consider a variation of the problem, where the large
hypergraphs~$H$ satisfy additional hereditary density conditions.
We present recent progress based on joint work with Reiher and
R\odl. In particular, we established a computer-free proof
of a recent result of Glebov, Kr{’a}l’, and Volec on the Tur’an-density
of the $3$-uniform hypergraph with three edges on four vertices
for hypergraphs that are hereditarily dense on large vertex sets.