Some Exact Results And New Asymptotics For Hypergraph Turán Numbers

Given a family F of r-graphs, let ex(n,F) be the maximum number of edges in an n vertex r-graph containing no member of F . Let C 4 denote the family of r-graphs with distinct edges A,B, C, D, such that A ∩ B = C ∩ D = ∅, A ∪ B = C ∪ D. For s1 ≤ · · · ≤ sr, let K(s1, . . . , sr) be the complete r-partite r-graph with parts of sizes s1, . . . , sr. Füredi conjectured over 15 years ago that ex(n, C 4 ) ≤ ( n 2 ) for n sufficiently large. We prove the weaker result ex(n, {C 4 ,K(3)(1, 2, 4)}) ≤ ( n 2 ) . Generalizing a well-known conjecture for the Turán number of bipartite graphs, we conjecture that ex(n, K(s1, . . . , sr)) = Θ(nr−1/s), where s = ∏r−1 i=1 si. We prove this conjecture when s1 = · · · = sr−2 = 1 and (i) sr−1 = 2, (ii) sr−1 = sr = 3, (iii) sr > (sr−1 − 1)!. In cases (i) and (ii), we determine the asymptotic value of ex(n, K(s1, . . . , sr)). We also provide an explicit construction giving ex(n, K(2, 2, 3)) > (1/6− o(1))n. This improves upon the previous best lower bound of Ω(n29/11) obtained by probabilistic methods. Several related open problems are also presented. ∗School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, [email protected] Research supported in part by the National Science Foundation under grant DMS-9970325 1991 Mathematics Subject Classification: 05B05, 05B07, 05B25, 05C35, 05C80, 05D05, 11T99