Asymptotics for the Turán number of Berge-$K_{2,t}$

Abstract

Let $F$ be a graph. A hypergraph is called Berge-$F$ if it can be obtained by replacing each edge of $F$ by a hyperedge containing it. Let $\mathcal{F}$ be a family of graphs. The Turán number of Berge-$\mathcal{F}$ is the maximum possible number of edges in an $r$-uniform hypergraph on $n$ vertices containing no Berge-$F$ as a subhypergraph (for every $F\in \mathcal{F}$) and is denoted by $e{x}_r(n,\mathcal{F})$. We determine the asymptotics for the Turán number of Berge-$K_{2,t}$ by showing $$e{x}_3(n,K_{2,t})=\frac{1}{6}(t-1{)}^{3/2}\cdot n^{3/2}(1+o(1))$$ for any given $t\ge 7$. We study the analogous question for linear hypergraphs and show that $$e{x}_3(n,\{C_2,K_{2,t}\})=\frac{1}{6}\sqrt{t-1}\cdot n^{3/2}(1+o_t(1)).$$ We also prove general upper and lower bounds on the Turán numbers of a class of graphs including $e{x}_r(n,K_{2,t})$, $e{x}_r(n,\{C_2,K_{2,t}\})$, and $e{x}_r(n,C_{2k})$ for $r\ge 3$. Our bounds improve results of Gerbner and Palmer, Füredi and Özkahya, Timmons, and provide a new proof of a result of Jiang and Ma.

arXiv comment25 pages; Writing improved based on the suggestions of the referees and mistakes corrected