Embedding tetrahedra into quasirandom hypergraphs

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 $d\mathrm{\mid }X\mathrm{\mid }\mathrm{\mid }P\mathrm{\mid }\pm \eta \mathrm{\mid }V{\mathrm{\mid }}^3$. We show that for any $\epsilon >0$ there exists $\eta >0$ such that every sufficiently large $(1\mathrm{/}2+\epsilon ,\eta )$-quasirandom hypergraph contains a tetrahedron, i.e., four vertices spanning all four hyperedges. A known random construction shows that the density $1\mathrm{/}2$ 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 $1\mathrm{/}2$, i.e., every large subset of vertices induces a hypergraph with density bigger than $1\mathrm{/}2$, contains a tetrahedron.