Talk information

Date: Sunday, January 26, 2025
Time: 10:10–11:00
Place: Schreiber 309
Speaker: Chaya Keller (Ariel University)
Title: An $(\aleph_0,k+2)$-Theorem for $k$-Transversals

Abstract:

A family $F$ of sets satisfies the $(p,q)$-property if among every $p$ members of $F$, some $q$ can be pierced by a single point. The celebrated $(p,q)$-theorem of Alon and Kleitman asserts that for any $p \geq q \geq d+1$, any family $F$ of compact convex sets in $R^d$ that satisfies the $(p,q)$-property can be pierced by a finite number $c(p,q,d)$ of points. A similar theorem with respect to piercing by $(d-1)$-dimensional flats, called $(d-1)$-transversals, was obtained by Alon and Kalai.

In this talk we prove the following result, which can be viewed as an $(\aleph_0,k+2)$-theorem with respect to $k$-transversals: Let $F$ be an infinite family of closed balls in $R^d$, and let $0 \leq k < d$. If among every $\aleph_0$ elements of $F$, some $k+2$ can be pierced by a $k$-dimensional flat, then $F$ can be pierced by a finite number of $k$-dimensional flats. Furthermore, the same holds for a wider class of “well-behaved” compact sets that do not have to be convex. This is the first $(p,q)$-theorem in which the assumption is weakened to an $(\infty,\cdot)$ assumption. Time permitting, we will discuss the relation between the existence of $(p,q)$-theorems for all sufficiently large $p$, and the existence of an $(\aleph_0,q)$-theorem. No background in infinitary combinatorics is required.

Joint work with Micha A. Perles.