On 3-uniform hypergraphs without a cycle of a given length

We study the maximum number of hyperedges in a 3-uniform hypergraph on n vertices that does not contain a Berge cycle of a given length l. In particular we prove that the upper bound for C2k+1-free hypergraphs is of the order O(kn), improving the upper bound of Győri and Lemons [10] by a factor of Θ(k). Similar bounds are shown for linear hypergraphs. 1. A generalization of the Turán problem Counting substructures is a central topic of extremal combinatorics. Given two (hyper)graphs G and H let N(G ;H) denote the number of subgraphs of G isomorphic to H. (Usually we consider a labelled host graph G). Note that N(G ;K2) = e(G), the number of edges of G. More generally, N(G ;H) is the maximum of N(G ;H) where G ∈ G, a class of graphs. In most cases, in Turán type problems, G is a set of n-vertex F-free graphs, where F is a collection of forbidden subgraphs. This maximum is denoted by N(n,F ;H). So N(n,F ;H) is the maximum number of copies of H in an F-free graph on n vertices. The Turán number ex(n,F) is defined as N(n,F ;K2). Let ex(m,n,F) be the maximum number edges in a bipartite graph with parts of order m and n vertices that do not contain any member of F . Cl is the family of all cycles of length at most l. For any graph G and any vertex x, we let t(G) and t(x) denote the number of triangles in G and the number of triangles containing x, respectively. Let tl(n) := N(n,Cl ;K3). Our starting point is the Bondy-Simonovits [3] theorem, ex(n,C2k) ≤ 100kn1+1/k . Recall two contemporary versions due to Pikhurko [15], Bukh and Z. Jiang [4], respectively, and a The research of the first author is supported in part by the Hungarian National Science Foundation OTKA 104343, by the European Research Council Advanced Investigators Grant 26719 and by the Simons Foundation grant 317487. This work was done while the first author visited the Department of Mathematics and Computer Science, Emory University, Atlanta, GA, USA. A major revision of the paper was done during a visit of the first author to the Institut Mittag-Leffler (Djursholm, Sweden). 2010 Mathematics Subject Classifications: 05C35, 05C65, 05D05.