### Reuniting chi-boundedness with polynomial chi-boundedness

James Davies#### BBC+G Seminar

on 9/21/23

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