On the Traces of Complete Hypergraphs

Let n ≥ t ≥ 3. Suppose that F is a family of t-element subsets of [n] satisfying |F| > ( n t−1 )t−1 + o(nt−1). Our main result implies that there exists a t-element subset S of [n] such that {E ∩ S : E ∈ F} contains all 2-element subsets of S, and this is asymptotically sharp. When t = 3, we prove the same result whenever |F| > ⌊ (n−1)2 4 ⌋ , which is sharp for all n ≥ 3. Our proofs use the Erdős-Rado Sunflower Lemma and recent results on hypergraph Turán numbers.