Extremal hypergraphs for certain forbidden traces

We consider two problems from the class of extremal problems, which we call "forbidden trace" problems, in which maximum-sized hypergraphs that do not contain a certain subhypergraph or class of subhypergraphs are studied. Let H = (V, E ) be a hypergraph with vertex set V and edge set E . For any X ⊆ V define the trace of H on X to be the hypergraph with vertex set X and edge set E&parl0;H&vbm0;X&parr0; = {lcub}E ∩ X : E ∈ H {rcub}. Given a hypergraph F we say that H&rarrr;F if there is a set X ⊂ V such that F is contained in H |X.; One line of research is to consider a multi-hypergraph F with a small number of vertices, a hypergraph H with n vertices such that H↛F and determine the maximum number of edges in H as a function of n. Asymptotically exact bounds have been found for the maximum number of edges in hypergraphs that do not contain F as a trace for all besides three of the hypergraphs F with three vertices. It is believed that the solution for the following hypergraph will lead to the solution of the remaining two cases. For any integer t ≥ 2 we define Ft to be the hypergraph with vertex set {lcub}1, 2, 3{rcub} whose edge set consists of t copies of each edge of the edges {lcub}1, 2{rcub}, {lcub}1, 3{rcub}, {lcub}2, 3{rcub}. Anstee, Griggs and Sali conjecture that the maximum number of edges in a hypergraph H with n vertices such that H↛Ft is O(n2); we show that it is at most O( n73 ).