Slalom numbers

TL;DR

This work develops a unified framework for slalom numbers, a broad family of cardinal invariants parameterized by ideals on $\omega$, via relational systems and Tukey-order methods. It connects localization/anti-localization phenomena to topological selection principles, deriving extensive identifications with classical invariants (e.g., $\mathfrak{d}=\mathfrak{sl}_t(\mathrm{Fin},\mathrm{Fin})$, $\mathfrak{b}=\mathfrak{sl}_e(\mathrm{Fin},\mathrm{Fin})$, $\mathfrak{p}=\mathfrak{sl}_e(\star,\mathrm{Fin})$) and establishing monotonicity, disjoint-sum behavior, and product/diagonal relations. The paper then develops forcing and coherent-system techniques to realize a rich landscape of constellations, notably showing how Cohen reals can produce many independent slalom-number values and how ccc models can separate multiple invariants. It concludes with a comprehensive set of open problems regarding equalities across ideal choices, equivalences of selection principles, and extending the forcing toolkit to generate further diverse configurations. Overall, it provides a powerful, versatile toolkit for analyzing a wide class of continuum invariants through slalom numbers and their connections to topology and set theory.

Abstract

The paper is an extensive and systematic study of cardinal invariants we call slalom numbers, describing the combinatorics of sequences of sets of natural numbers. Our general approach, based on relational systems, covers many such cardinal characteristics, including localization and anti-localization cardinals. We show that most of the slalom numbers are connected to topological selection principles, in particular, we obtain the representation of the uniformity of meager and the cofinality of measure. Considering instances of slalom numbers parametrized by ideals on natural numbers, we focus on monotonicity properties with respect to ideal orderings and computational formulas for the disjoint sum of ideals. Hence, we get such formulas for several pseudo-intersection numbers as well as for the bounding and dominating numbers parametrized with ideals. Based on the effect of adding a Cohen real, we get many consistent constellations of different values of slalom numbers.

Figures (18)

• Figure 1: Diagram of inequalities between slalom numbers. An arrow denotes that ZFC proves $\leq$.
• Figure 2: Relations among particular cases of slalom numbers.
• Figure 3: Further connections between slalom numbers
• Figure 4: Critical cardinality of some selection principles.
• Figure 5: Effect of adding $\lambda$ many Cohen reals.

Theorems & Definitions (152)

• Theorem 1: \ref{['G2']}
• Theorem 2
• Theorem 3: \ref{['L6.2']}, \ref{['L6.3']}, and \ref{['L6.4']}
• Theorem 4
• Theorem 5: \ref{['aplcohen']}
• Example 2.1
• Corollary 2.3
• Definition 2.4
• Definition 3.1
• Lemma 3.2