Finite k-Transversals of Infinite Families of Fat Convex Sets

TL;DR

This paper extends the theory of piercing convex sets with flats in Euclidean space, introducing new infinite and colorful variants of the $(p,q)$-theorem, and explores the relationship between finite and infinite cases.

Contribution

It proves an infinite $(p,q)$-theorem for fat convex sets with $k$-flats and develops a framework for colorful variants, highlighting differences between finite and infinite theorems.

Findings

Established an infinite $(p,q)$-theorem for fat convex sets and $k$-flats.

Developed a new framework for colorful variants of $(p,q)$-theorems.

Showed that infinite theorems do not imply finite counterparts.

Abstract

We prove an infinite $(p,q)$-theorem for piercing fat compact convex sets in $\RR^d$ with $k$-flats. Additionally, we develop a new framework through which infinite $(p,q)$-theorems concerning compact sets and $k$-flats can be extended to their 'colorful' variants. Further, we show that the existence of an infinite $(p,q)$-theorem does not necessarily imply the existence of the corresponding finite $(p,q)$-theorem.