A hypergraph extension of Turán's theorem

Fix l ≥ r ≥ 2. Let H l+1 be the r-uniform hypergraph obtained from the complete graph Kl+1 by enlarging each edge with a new set of r− 2 vertices. Thus H l+1 has (r− 2) ( l+1 2 ) + l +1 vertices and ( l+1 2 ) edges. We prove that the maximum number of edges in an n-vertex r-uniform hypergraph containing no copy of H l+1 is (l)r lr ( n r ) + o(n) as n → ∞. This is the first infinite family of irreducible r-uniform hypergraphs for each odd r > 2 whose Turán density is determined. Along the way we give three proofs of a hypergraph generalization of Turán’s theorem. We also prove a stability theorem for hypergraphs, analogous to the Simonovits stability theorem for complete graphs.