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

Description of video

Date: 3/25/22
Speaker :McGinnis Daniel

A family of sets is said to have the (p,q)-property if for every p sets, q of them have a common point. It was shown by Alon and Kleitman that if F is a finite family of convex sets in Rd and qd+1, then there is come constant cd(p,q) number of points that pierces each set in F. A problem of interest is to improve the bounds on the numbers cd(p,q). Here, we show that c2(4,3)9, which improves the previous upper bound of 13 by Gyárfás, Kleitman, and Tóth. The proof combines a topological argument and a geometric analysis.