# 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

## Keywords

A family of sets is said to have the $\left(p,q\right)$-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 ${\mathbb{R}}^{d}$ and $q\ge d+1$, then there is come constant ${c}_{d}\left(p,q\right)$ number of points that pierces each set in $F$. A problem of interest is to improve the bounds on the numbers ${c}_{d}\left(p,q\right)$. Here, we show that ${c}_{2}\left(4,3\right)\le 9$, which improves the previous upper bound of 13 by Gyárfás, Kleitman, and Tóth. The proof combines a topological argument and a geometric analysis.

## Related videos

Zvavitch Artem

on 4/29/22

Naszódi Márton

on 2/18/22
01:10:00

### Jacques Verstraete: Extremal problems for geometric hypergraphs

Verstraete Jacques

on 11/20/20