A hypergraph extension of Turán's theorem

Abstract

Fix l ⩾ r ⩾ 2 . Let H l + 1 ( r ) be the r -uniform hypergraph obtained from the complete graph K l + 1 by enlarging each edge with a set of r - 2 new vertices. Thus H l + 1 ( r ) 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 ( r ) is ( l ) r l r n r + o ( n r ) 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.