On intersecting straight line systems

Abstract

An intersecting r-uniform straight line system is an intersecting linear system whose lines consist of r points on straight line segment of ℝ² and any two lines share a point (are intersecting). On the other hand, an intersecting r-segment system is an intersecting linear system whose lines consist of r consecutive integer points on a line segment in ℤ² and any two lines share a point. It is not hard to see that an intersecting r-segment system is an intersecting r-uniform straight line system.

Recently, Oliveros et al. (2020) proved any intersecting r-segment system has transversal number at most r - 1. In this paper, we improve such result given by Oliveros et al., we prove that any intersecting r-uniform straight line system has a transversal number at most ν₂ - 1, where ν₂ is the maximum cardinality of a subset of lines R of the straight line system such that every triplet of different elements of R doesn’t have a common point.

Subject Classification:

• 05C65
• 05C69

Keywords:

• Linear systems
• Straight line systems
• Segment systems
• Transversal number
• 2-packing number