Asyrnptotics for the Turan number of Berge-K-2,K-t

Let F be a graph. A hypergraph is called Berge-F if it can be obtained by replacing each edge in F by a hyperedge containing it. Let F be a family of graphs. The Turan number of the family Berge-F is the maximum possible number of edges in an r-uniform hypergraph on n vertices containing no Berge-F as a subhypergraph (for every F is an element of F) and is denoted by ex(r)(n, F). We determine the asymptotics for the Turan number of Berge-K-2,K-t, by showing ex(3)(n, K-2,K-t) = (1 + o(1))1/6 (t - 1)(3/2) . n(3/2) for any given t >= 7. We study the analogous question for linear hypergraphs and show ex(3)(n,{C-2, K-2,K-t}) = (1 + o(t)(1))1/6 root t - 1. n(3/2). We also prove general upper and lower bounds on the Turan numbers of a class of graphs including ex(r)(n, K-2,K-t), ex(r)(n, {C-2, K-2,K-t}), and ex,(n, C-2k) for r >= 3. Our bounds improve the results of Gerbner and Palmer [18], Fiiredi and Ozkahya [15], Timmons [37], and provide a new proof of a result of Jiang and Ma [26]. (C) 2019 Elsevier Inc. All rights reserved.