Cichon's Diagram for uncountable cardinals

TL;DR

This paper extends Cichon's diagram to uncountable regular cardinals, generalizing known results and establishing new independence results for cardinal invariants on generalized Cantor and Baire spaces.

Contribution

It develops a generalized Cichon's diagram for uncountable cardinals, proving ZFC results and independence results for cardinal invariants in this broader context.

Findings

Generalization of ZFC results for cardinal invariants to uncountable regular cardinals

Establishment of a natural generalization of the Bartoszynski-Raisonnier-Stern Theorem

Independence results showing strict inequalities between invariants

Abstract

We develop a version of Cichon's diagram for cardinal invariants on the generalized Cantor space 2^kappa or the generalized Baire space kappa^kappa where kappa is an uncountable regular cardinal. For strongly inaccessible kappa, many of the ZFC-results about the order relationship of the cardinal invariants which hold for omega generalize; for example we obtain a natural generalization of the Bartoszynski-Raisonnier-Stern Theorem. We also prove a number of independence results, both with <kappa-support iterations and kappa-support iterations and products, showing that we consistently have strict inequality between some of the cardinal invariants.