Asymptotics for the Turán number of Berge-K2, t

Abstract 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 ∈ F ) and is denoted by e x r ( n , F ) . We determine the asymptotics for the Turan number of Berge- K 2 , t by showing e x 3 ( n , K 2 , 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 e x 3 ( n , { C 2 , K 2 , t } ) = ( 1 + o t ( 1 ) ) 1 6 t − 1 ⋅ n 3 / 2 . We also prove general upper and lower bounds on the Turan numbers of a class of graphs including e x r ( n , K 2 , t ) , e x r ( n , { C 2 , K 2 , t } ) , and e x r ( n , C 2 k ) for r ≥ 3 . Our bounds improve the results of Gerbner and Palmer [18] , Furedi and Ozkahya [15] , Timmons [37] , and provide a new proof of a result of Jiang and Ma [26] .