Rosenthal families, pavings and generic cardinal invariants

TL;DR

This paper characterizes Rosenthal families of infinite subsets of natural numbers, showing their minimal size equals the reaping cardinal, and explores related cardinal invariants using paving lemmas and operator theory.

Contribution

It proves that every ultrafilter is a Rosenthal family and establishes the minimal size of such families as the reaping cardinal, connecting combinatorial and operator-theoretic methods.

Findings

Every ultrafilter is a Rosenthal family.

Minimal size of Rosenthal families equals the reaping cardinal.

Connections with free set results and linear operators on c_0.

Abstract

Following D. Sobota we call a family $\mathcal F$ of infinite subsets of $\mathbb N$ a Rosenthal family if it can replace the family of all infinite subsets of $\mathbb N$ in classical Rosenthal's Lemma concerning sequences of measures on pairwise disjoint sets. We resolve two problems on Rosenthal families: every ultrafilter is a Rosenthal family and the minimal size of a Rosenthal family is exactly equal to the reaping cardinal $\mathfrak r$. This is achieved through analyzing nowhere reaping families of subsets of $\mathbb N$ and through applying a paving lemma which is a consequence of a paving lemma concerning linear operators on $\ell_1^n$ due to Bourgain. We use connections of the above results with free set results for functions on $\mathbb N$ and with linear operators on $c_0$ to determine the values of several other derived cardinal invariants.