On a problem of Henning and Yeo about the transversal number of uniform linear systems whose 2-packing number is fixed

Abstract

A linear system is a pair $(P,\mathcal{L})$ where $\mathcal{L}$ is a family of subsets on a ground finite set $P$ such that $|l\cap l^\prime|\leq 1$, for every $l,l^\prime \in \mathcal{L}$. If all elements of $\mathcal{L}$ of a linear system $(P,\mathcal{L})$, then the linear system is called $r$-uniform linear system. The transversal number of a linear system $(P,\mathcal{L})$, $\tau(P,\mathcal{L})$, is the minimum cardinality of a subset $\hat{P}\subseteq P$ satisfying $l\cap\hat{P}\neq\emptyset$, for every $l\in\mathcal{L}$. The 2-packing number of a linear system $(P,\mathcal{L})$, $\nu_2(P,\mathcal{L})$, is the maximum cardinality of a subset $R\subseteq\mathcal{L}$ such that, any three elements of $R$ don't have a common point (are triplewise disjoint), that is, if three elements are chosen in $R$, then they are not incidents in a common point. For $r\geq2$, let $(P,\mathcal{L})$ be an $r$-uniform linear system. In "{\sc M. A. Henning and A. Yeo:} {\it Hypergraphs with large transversal number,} Discrete Math. {\bf 313} (2013), no. 8, 959--966." Henning and Yeo state the following question: Is it true that if $(P,\mathcal{L})$ is an $r$-uniform linear system then $\tau(P,\mathcal{L})\leq\displaystyle\frac{|P|+|\mathcal{L}|}{r+1}$ holds for all $r\geq2$?. In this note, we give some results of $r$-uniform linear systems, whose 2-packing number is fixed, satisfying the inequality.