Asymptotic Turán number for linear 5-cycle in 3-uniform linear hypergraphs

Abstract

An r-uniform hypergraph is linear if every two edges intersect in at most one vertex. Given a family of r-uniform hypergraphs F, the linear Turán number exrlin(n,F) is the maximum number of edges of a linear r-uniform hypergraph on n vertices that does not contain any member of F as a subhypergraph. For each k≥3, the linear k-cycle Ck is the 3-uniform linear hypergraph with edges h1,…,hk such that for every 1≤i≤k−1, |hi∩hi+1|=1,|hk∩h1|=1 and hi∩hj=∅ for all other pairs {i,j},i≠j.It is proved by Collier-Cartaino, Graber, Jiang [3] and Ergemlidze, Győri, Methuku [4] that ex3lin(n,C5)=Θ(n3/2). In this paper, we strengthen their results by proving thatex3lin(n,C5)=133⋅n3/2+O(n).