Some Remarks on π

We investigate extremal problems for hypergraphs satisfying the following density condition. A 3-uniform hypergraph H = (V, E) is (d, η,) -dense if for any two subsets of pairs P, Q V ×V the number of pairs ((x, y), (x, z)) P × Q with {x, y, z} E is at least d |K (P, Q)| - η |V|3, where K (P, Q) denotes the set of pairs in P × Q of the form ((x, y), (x, z)). For a given 3-uniform hypergraph F we are interested in the infimum d ≥ 0 such that for sufficiently small η every sufficiently large (d, η,) -dense hypergraph H contains a copy of F and this infimum will be denoted by π (F). We present a few results for the case when F = K( ³) _k is a complete 3-uniform hypergraph on k vertices. It will be shown that, which is sharp for r = 2, 3, 4, where the lower bound for r = 4 is based on a result of Chung and Graham [Edge-colored complete graphs with precisely colored subgraphs, Combinatorica 3, 3-4 (1983), 315-324].