A family of convex sets in the plane satisfying the $(4,3)$-property can be pierced by nine points

TL;DR

This paper proves that any finite family of convex sets in the plane with the (4,3)-property can be intersected by only nine points, improving previous bounds from thirteen.

Contribution

The authors establish a new, tighter bound of nine piercing points for convex sets satisfying the (4,3)-property, advancing geometric piercing theory.

Findings

Reduced the piercing number from 13 to 9 for (4,3)-property families

Established a new bound applicable to finite convex set families in the plane

Improved understanding of geometric intersection properties

Abstract

We prove that every finite family of convex sets in the plane satisfying the $(4,3)$-property can be pierced by $9$ points. This improves the bound of $13$ proved by Gy\'arf\'as, Kleitman, and T\'oth in 2001.