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 $|lcap l^prime|leq 1$, for every $l,l^prime in mathcal{L}$. The elements of $P$ and $mathcal{L}$ are called points and lines, respectively, and the linear system is called intersecting if any pair of lines intersect in exactly one point. A subset $T$ of points of $P$ is a transversal of $(P,mathcal{L})$ if $T$ intersects any line, and the transversal number, $tau(P,mathcal{L})$, is the minimum order of a transversal. On the other hand, a 2-packing set of a linear system $(P,mathcal{L})$ is a set $R$ of lines, such that any three of them have a common point, then the 2-packing number of $(P,mathcal{L})$, $ u_2(P,mathcal{L})$, is the size of a maximum 2-packing set. It is known that the transversal number $tau(P,mathcal{L})$ is bounded above by a quadratic function of $ u_2(P,mathcal{L})$. An open problem is to haracterize the families of linear systems which satisfies $tau(P,mathcal{L})leq lambda u_2(P,mathcal{L})$, for some $lambdageq1$. In this paper, we give an infinite family of linear systems $(P,mathcal{L})$ which satisfies $tau(P,mathcal{L})= u_2(P,mathcal{L})$ with smallest possible cardinality of $mathcal{L}$, as well as some properties of $r$-uniform intersecting linear systems $(P,mathcal{L})$, such that $tau(P,mathcal{L})= u_2(P,mathcal{L})=r$. Moreover, we state a characterization of $4$-uniform intersecting linear systems $(P,mathcal{L})$ with $tau(P,mathcal{L})= u_2(P,mathcal{L})=4$.
تحميل البحث