Higher Dimensional Cardinal Characteristics for Sets of Functions II

TL;DR

This paper investigates higher dimensional cardinal characteristics for sets of functions, revealing that bounding numbers can be less than the continuum while dominating numbers cannot, and explores their behavior in various models of set theory.

Contribution

It provides new results on the values and relationships of higher dimensional bounding and dominating numbers, including their behavior in different set-theoretic models.

Findings

Bounding numbers can be strictly less than the continuum.

Dominating numbers cannot be less than the continuum.

In many models, bounding numbers are computed explicitly.

Abstract

We study the values of the higher dimensional cardinal characteristics for sets of functions $f:\omega^\omega \to \omega^\omega$ introduced by the second author. We prove that while the bounding numbers for these cardinals can be strictly less than the continuum, the dominating numbers cannot. We compute the bounding numbers for the higher dimensional relations in many well known models of $\neg\mathsf{CH}$ such as the Cohen, random and Sacks models and, as a byproduct show that, with possibly one exception, for the bounding numbers there are no $\mathsf{ZFC}$ relations between them beyond those in the higher dimensional Cicho\'{n} diagram. In the case of the dominating numbers we show that in fact they collapse in the sense that modding out by the ideal does not change their values. Moreover, they are closely related to the dominating numbers $\mathfrak{d}^\lambda_\kappa$.