Asymptotics for Turán numbers of cycles in 3-uniform linear hypergraphs

Abstract

Let F be a family of 3-uniform linear hypergraphs. The linear Turán number of F is the maximum possible number of edges in a 3-uniform linear hypergraph on n vertices which contains no member of F as a subhypergraph.In this paper we show that the linear Turán number of the five cycle C5 (in the Berge sense) is 133n3/2 asymptotically. We also show that the linear Turán number of the four cycle C4 and {C3,C4} are equal asymptotically, which is a strengthening of a theorem of Lazebnik and Verstraëte [16].We establish a connection between the linear Turán number of the linear cycle of length 2k+1 and the extremal number of edges in a graph of girth more than 2k−2. Combining our result and a theorem of Collier-Cartaino, Graber and Jiang [8], we obtain that the linear Turán number of the linear cycle of length 2k+1 is Θ(n1+1k) for k=2,3,4,6.