Forbidding Complete Hypergraphs as Traces

Let 2 ≤q ≤min{p, t − 1} be fixed and n → ∞. Suppose that is a p-uniform hypergraph on n vertices that contains no complete q-uniform hypergraph on t vertices as a trace. We determine the asymptotic maximum size of in many cases. For example, when q = 2 and p∈{t, t + 1}, the maximum is , and when p = t = 3, it is for all n≥ 3. Our proofs use the Kruskal-Katona theorem, an extension of the sunflower lemma due to Füredi, and recent results on hypergraph Turán numbers. Research supported in part by NSF grants DMS-0400812 and an Alfred P. Sloan Research Fellowship. Research supported in part by NSA grant H98230-06-1-0140. Part of the research conducted while his working at University of Illinois at Chicago as a NSF VIGRE postdot.