Forbidding Complete Hypergraphs as Traces

Let 2 ≤ q ≤ min{p, t − 1} be fixed and n → ∞. Suppose that F 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 F in many cases. For example, when q = 2 and p ∈ {t, t+ 1}, the maximum is ( n t−1 ) t−1 + o(nt−1), and when p = t = 3, it is b (n−1) 2 4 c 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. Email: [email protected] †Research supported in part by NSA grant H98230-06-1-0140. Part of the research conducted while working at University of Illinois at Chicago as a NSF VIGRE postdot. Email: [email protected]