Embedding tetrahedra into quasirandom hypergraphs

We investigate extremal problems for quasirandom hypergraphs. We say that a 3-uniform hypergraph H “ pV,Eq is pd, η, q-quasirandom if for any subset X Ď V and every set of pairs P Ď V ˆV the number of pairs px, py, zqq P XˆP with tx, y, zu being a hyperedge of H is in the interval d |X| |P | ̆ η |V |. We show that for any ε ą 0 there exists η ą 0 such that every sufficiently large p1{2` ε, η, q-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 Erdős, whether every weakly quasirandom 3-uniform hypergraph H is 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.