On transversal and 2-packing numbers in uniform linear systems

Abstract

A linear system is a pair where is a family of subsets on a ground finite set , such that , for every . The elements of and 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 of points of is a transversal of if intersects any line, and the transversal number, , is the minimum order of a transversal. On the other hand, a 2-packing set of a linear system is a set of lines, such that any three of them have a common point, then the 2-packing number of , , is the size of a maximum 2-packing set. It is known that the transversal number is bounded above by a quadratic function of . An open problem is to characterize the families of linear systems which satisfies , for some . In this paper, we give an infinite family of linear systems which satisfies with smallest possible cardinality of , as well as some properties of -uniform intersecting linear systems , such that . Moreover, we state a characterization of 4-uniform intersecting linear systems with .

Keywords

• linear systems
• transversal number
• 2-packing number
• finite projective plane