A hypergraph extension of Turán's theorem

Abstract

Fix l ≥ r ≥ 2. Let Hl+1(r) be the r-uniform hypergraph obtained from the complete graph Kl+1 by enlarging each edge with a set of r - 2 new vertices. Thus Hl+1(r) has (r - 2) (2l+1) + l + 1 vertices and (2l+1) edges. We prove that the maximum number of edges in an n-vertex r-uniform hypergraph containing no copy of Hl+1(r) is (l)r/lr (rn)+o(nr) 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. © 2005 Published by Elsevier Inc.