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

For $r\geq2$, let $(P,\mathcal{L})$ be an $r$-uniform linear system. The transversal number $\tau(P,\mathcal{L})$ of $(P,\mathcal{L})$ is the minimum number of points that intersect every line of $(P,\mathcal{L})$. The 2-packing number $\nu_2(P,\mathcal{L})$ of $(P,\mathcal{L})$ is the maximum number of lines such that the intersection of any three of them is empty. In [Discrete Math. 313 (2013), 959--966] Henning and Yeo posed the following question: Is it true that if $(P,\mathcal{L})$ is a $r$-uniform linear system then $\tau(P,\mathcal{L})\leq\displaystyle\frac{|P|+|\mathcal{L}|}{r+1}$ holds for all $k\geq2$?. In this paper, some results about of $r$-uniform linear systems whose 2-packing number is fixed which satisfies the inequality are given

Transversal numbers over subsets of linear spaces

Let $M$ be a subset of $\mathbb{R}^k$. It is an important question in the theory of linear inequalities to estimate the minimal number $h=h(M)$ such that every system of linear inequalities which is infeasible over $M$ has a subsystem of at most $h$ inequalities which is already infeasible over $M.$ This number $h(M)$ is said to be the Helly number of $M.$ In view of Helly's theorem, $h(\mathbb{R}^n)=n+1$ and, by the theorem due to Doignon, Bell and Scarf, $h(\mathbb{Z}^d)=2^d.$ We give a common extension of these equalities showing that $h(\mathbb{R}^n \times \mathbb{Z}^d) = (n+1) 2^d.$ We show that the fractional Helly number of the space $M \subseteq \mathbb{R}^d$ (with the convexity structure induced by $\mathbb{R}^d$) is at most $d+1$ as long as $h(M)$ is finite. Finally we give estimates for the Radon number of mixed integer spaces