Revisiting $\mathfrak b$ and $\mathfrak d$ through Interval Structures

TL;DR

This paper explores interval-based relational systems related to classical set-theoretic invariants, showing that universal variants preserve these invariants while existential variants reverse them across various settings.

Contribution

It introduces natural interval-type generalizations of bounding and dominating numbers and demonstrates their robustness and systematic reversal in different variants.

Findings

Universal variants preserve classical invariants $rak b$ and $rak d$.

Existential variants reverse the invariants, making bounding number equal to $rak d$ and vice versa.

The invariants coincide with classical ones across multiple settings such as discrete, colored, and measure-theoretic.

Abstract

We investigate a family of relational systems arising from interval partitions of $\omega$, inspired by Vojt\'a\v{s}'s characterization of the bounding and dominating numbers. By varying the underlying asymptotic quantifiers and interval constraints, we obtain several natural interval-type generalizations. We show that the universal variants are remarkably robust: in all the discrete, colored, restricted, bounded, and measure-theoretic settings considered here, the associated bounding and dominating numbers coincide with the classical invariants $\mathfrak b$ and $\mathfrak d$. In contrast, the existential variants systematically reverse these invariants, yielding that the bounding number coincides with $\mathfrak d$ and the dominating number coincides with $\mathfrak b$.