  We investigate extremal problems for quasirandom hypergraphs. We say that a
$3$-uniform hypergraph $H=(V,E)$ is $(d,\eta)$-quasirandom if for any subset
$X\subseteq V$ and every set of pairs $P\subseteq V\times V$ the number of
pairs $(x,(y,z))\in X\times P$ with $\{x,y,z\}$ being a hyperedge of $H$ is in
the interval $d|X||P|\pm\eta|V|^3$. We show that for any $\varepsilon>0$ there
exists $\eta>0$ such that every sufficiently large
$(1/2+\varepsilon,\eta)$-quasirandom hypergraph contains a tetrahedron, i.e.,
four vertices spanning all four hyperedges. A known random construction shows
that the density $1/2$ is best possible. This result is closely related to a
question of Erd\H{o}s, whether every weakly quasirandom $3$-uniform hypergraph
$H$ with density bigger than $1/2$, i.e., every large subset of vertices
induces a hypergraph with density bigger than $1/2$, contains a tetrahedron.
