  For a $k$-uniform hypergraph $F$ let $\textrm{ex}(n,F)$ be the maximum number
of edges of a $k$-uniform $n$-vertex hypergraph $H$ which contains no copy of
$F$. Determining or estimating $\textrm{ex}(n,F)$ is a classical and central
problem in extremal combinatorics. While for $k=2$ this problem is well
understood, due to the work of Tur\'an and of Erd\H{o}s and Stone, only very
little is known for $k$-uniform hypergraphs for $k>2$. We focus on the case
when $F$ is a $k$-uniform hypergraph with three edges on $k+1$ vertices.
Already this very innocent (and maybe somewhat particular looking) problem is
still wide open even for $k=3$.
  We consider a variant of the problem where the large hypergraph $H$ enjoys
additional hereditary density conditions. Questions of this type were suggested
by Erd\H os and S\'os about 30 years ago. We show that every $k$-uniform
hypergraph $H$ with density $>2^{1-k}$ with respect to every large collections
of $k$-cliques induced by sets of $(k-2)$-tuples contains a copy of $F$. The
required density $2^{1-k}$ is best possible as higher order tournament
constructions show.
  Our result can be viewed as a common generalisation of the first extremal
result in graph theory due to Mantel (when $k=2$ and the hereditary density
condition reduces to a normal density condition) and a recent result of Glebov,
Kr\'al', and Volec (when $k=3$ and large subsets of vertices of $H$ induce a
subhypergraph of density $>1/4$). Our proof for arbitrary $k\geq 2$ utilises
the regularity method for hypergraphs.
