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 Turán 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 exr(n,F).We determine the asymptotics for the Turán number of Berge-K2,t by showingex3(n,K2,t)=(1+o(1))16(t−1)3/2⋅n3/2 for any given t≥7. We study the analogous question for linear hypergraphs and showex3(n,{C2,K2,t})=(1+ot(1))16t−1⋅n3/2.We also prove general upper and lower bounds on the Turán numbers of a class of graphs including exr(n,K2,t), exr(n,{C2,K2,t}), and exr(n,C2k) for r≥3. Our bounds improve the results of Gerbner and Palmer [18], Füredi and Özkahya [15], Timmons [37], and provide a new proof of a result of Jiang and Ma [26].