Cardinal Characteristics on Bounded Generalised Baire Spaces

TL;DR

This paper explores four families of cardinal characteristics on subspaces of the generalized Baire space, analyzing their relationships, non-triviality conditions, and consistency results in the context of large cardinals.

Contribution

It introduces bounded versions of key cardinal characteristics on generalized Baire spaces and investigates their interactions and consistency properties.

Findings

Identified conditions for non-triviality of the bounded cardinals.

Established relationships among the bounded cardinals and classical ones.

Proved the consistency of distinct cardinal configurations based on parameter choices.

Abstract

We will give an overview of four families of cardinal characteristics defined on subspaces $\prod_{\alpha\in\kappa}b(\alpha)$ of the generalised Baire space ${}^\kappa\kappa$, where $\kappa$ is strongly inaccessible and $b\in{}^\kappa\kappa$. The considered families are bounded versions of the dominating, eventual difference, localisation and anti-localisation numbers, and their dual cardinals. We investigate parameters for which these cardinals are non-trivial and how the cardinals relate to each other and to other cardinals of the generalised Cicho\'n diagram. Finally we prove that different choices of parameters may lead to consistently distinct cardinals.