Rothberger gaps in fragmented ideals

TL;DR

This paper studies the Rothberger number of fragmented ideals, revealing that it can vary widely, including being countably infinite or larger than measure additivity, and shows the consistency of many such different numbers.

Contribution

It introduces the concept of Rothberger numbers for fragmented ideals and demonstrates their diverse possible values, including the consistency of infinitely many distinct Rothberger numbers.

Findings

Rothberger number is $eth_1$ for the linear growth ideal.

Rothberger number exceeds measure additivity for the polynomial growth ideal.

It is consistent that there are continuum many different Rothberger numbers for fragmented ideals.

Abstract

The~\emph{Rothberger number} $\mathfrak{b} (\mathcal{I})$ of a definable ideal $\mathcal{I}$ on $\omega$ is the least cardinal $\kappa$ such that there exists a Rothberger gap of type $(\omega,\kappa)$ in the quotient algebra $\mathcal{P} (\omega) / \mathcal{I}$. We investigate $\mathfrak{b} (\mathcal{I})$ for a subclass of the $F_\sigma$ ideals, the fragmented ideals, and prove that for some of these ideals, like the linear growth ideal, the Rothberger number is $\aleph_1$ while for others, like the polynomial growth ideal, it is above the additivity of measure. We also show that it is consistent that there are infinitely many (even continuum many) different Rothberger numbers associated with fragmented ideals.