A hypergraph extension of the bipartite Turلn problem

Let t,n be integers with n⩾3t. For t⩾3, we prove that in any family of at least t4n2 triples from an n-element set X, there exist 2t triples A1,B1,A2,B2,…,At,Bt and distinct elements a,b∈X such that Ai∩Aj={a} and Bi∩Bj={b}, for all i≠j, andAi∩Bj=Ai−{a}=Bj−{b}for i=j,∅for i≠j.When t=2, we improve the upper bound t4n2 to 3n2+6n. This improves upon the previous best known bound of 3.5n2 due to Füredi. Some results concerning more general configurations of triples are also presented.