On $r$-uniform linear hypergraphs with no Berge-$K_{2,t}$

Abstract

Let $\mathcal{F}$ be an $r$-uniform hypergraph and $G$ be a multigraph. The hypergraph $\mathcal{F}$ is a Berge-$G$ if there is a bijection $f:E(G)\to E(\mathcal{F})$ such that $e\subseteq f(e)$ for each $e\in E(G)$. Given a family of multigraphs $\mathcal{G}$, a hypergraph $\mathcal{H}$ is said to be $\mathcal{G}$-free if for each $G\in \mathcal{G}$, $\mathcal{H}$ does not contain a subhypergraph that is isomorphic to a Berge-$G$. We prove bounds on the maximum number of edges in an $r$-uniform linear hypergraph that is $K_{2,t}$-free. We also determine an asymptotic formula for the maximum number of edges in a linear 3-uniform 3-partite hypergraph that is $\{C_3,K_{2,3}\}$-free.

arXiv comment15 pages; the statement of Theorem 1.4 has been corrected