A hypergraph extension of Turán's theorem

Fix l⩾r⩾2l⩾r⩾2. Let Hl+1(r) be the rr-uniform hypergraph obtained from the complete graph Kl+1Kl+1 by enlarging each edge with a set of r-2r-2 new vertices. Thus Hl+1(r) has (r-2)l+12+l+1 vertices and l+12 edges. We prove that the maximum number of edges in an nn-vertex rr-uniform hypergraph containing no copy of Hl+1(r) is(l)rlrnr+o(nr)as n→∞n→∞. This is the first infinite family of irreducible rr-uniform hypergraphs for each odd r>2r>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.