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

TL;DR

This paper investigates the maximum edge counts in r-uniform linear hypergraphs avoiding certain Berge subgraphs, providing bounds and asymptotic formulas for specific cases involving complete bipartite graphs and cycles.

Contribution

It establishes new bounds on edges in Berge-$K_{2,t}$-free hypergraphs and derives an asymptotic formula for a specific 3-uniform hypergraph class.

Findings

Bounds on edges in Berge-$K_{2,t}$-free hypergraphs.

Asymptotic formula for linear 3-uniform 3-partite hypergraphs avoiding $C_3$ and $K_{2,3}$.

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 $f: E(G) \rightarrow E( \mathcal{F} )$ 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.