Finite k-Transversals of Infinite Families of Fat Convex Sets

We prove an infinite $(p,q)$-theorem for piercing fat compact convex sets in ${\text{RR}}^d$ 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.