Slalom numbers

TL;DR

This paper systematically studies slalom numbers, a class of cardinal invariants related to combinatorics of sequences of natural number sets, linking them to topological principles and analyzing their behavior under various set-theoretic assumptions.

Contribution

It introduces a unified framework for slalom numbers using relational systems, connecting them to topological selection principles and providing formulas and consistency results.

Findings

Most slalom numbers relate to topological selection principles.

Formulas for disjoint sums of ideals and pseudo-intersection numbers are established.

Consistent variations of slalom numbers are obtained via Cohen real addition.

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.