Supersaturation For Ramsey-Turán Problems

For an l-graph G, the Turán number ex(n,G) is the maximum number of edges in an n-vertex l-graph H containing no copy of G. The limit π(G) = limn→∞ ex (n,G)/ ( n l ) is known to exist [8]. The Ramsey-Turán density ρ(G) is defined similarly to π(G) except that we restrict to only those H with independence number o(n). A result of Erdős and Sós [3] states that π(G) = ρ(G) as long as for every edge E of G there is another edge E′ of G for which |E ∩E′| ≥ 2. Therefore a natural question is whether there exists G for which ρ(G) < π(G). Another variant ρ̃(G) proposed in [3] requires the stronger condition that every set of vertices of H of size at least 2n (0 < 2 < 1) has density bounded below by some threshold. By definition, ρ̃(G) ≤ ρ(G) ≤ π(G) for every G. However, even ρ̃(G) < π(G) is not known for very many l-graphs G when l > 2. We prove the existence of a phenomenon similar to supersaturation for Turán problems for hypergraphs. As a consequence, we construct, for each l ≥ 3, infinitely many l-graphs G for which 0 < ρ̃(G) < π(G). We also prove that the 3-graph G with triples 12a, 12b, 12c, 13a, 13b, 13c, 23a, 23b, 23c, abc, satisfies 0 < ρ(G) < π(G). The existence of a hypergraph H satisfying 0 < ρ(H) < π(H) was conjectured by Erdős and Sós [3], proved by Frankl and Rödl [6], and later by Sidorenko [14]. Our short proof is based on different ideas and is simpler than these earlier proofs. ∗Department of Mathematics, Statistics, and Computer Science, University of Illinois, 851 S. Morgan Street, Chicago, IL 60607-7045; research supported in part by the National Science Foundation under grants DMS-9970325 and DMS-0400812, and an Alfred P. Sloan Research Fellowship †Department of Mathematics and Computer Science, Emory University, Atlanta, GA 30322, USA; research supported in part by the National Science Foundation under grants DMS-0071261 and DMS-0300529 2000 Mathematics Subject Classification: 05C35, 05C65, 05D05