Variants of Baumgartner's Axiom for Lipschitz Functions on Baire and Cantor Space

TL;DR

The paper introduces Lipschitz variants of Baumgartner's axiom BA for the Polish spaces ω^ω and 2^ω, establishing consistency via ccc forcing from CH and under PFA. It proves a key implication: BA_Lip(ω^ω) implies BA_Lip(2^ω), and shows that strong Lipschitz versions push all cardinals in Cichoń's diagram above aleph_1, with corresponding consequences for the bounding number and null-additivity. The additivity of the null ideal is shown to exceed aleph_1 under the strong Lipschitz axioms, with separate treatments for ω^ω and 2^ω leveraging Bartoszyński’s and Shelah’s characterizations. Finally, the work demonstrates that these Lipschitz axioms need not follow from large fragments of MA, via an iteration argument preserving Lipschitz-avoiding sets, and outlines open questions about the full landscape of implications and equivalences among the Lipschitz variants.

Abstract

We consider several variants of Baumgartner's axiom for $\aleph_1$-dense sets defined on the Baire and Cantor spaces in terms of Lipschitz functions with respect to the usual metric. A variation of Baumgartner's original argument shows that these variants are consistent. However, unlike in the case of the classical $\mathsf{BA}$, we are able to give many applications for which the corresponding fact for linear orders is open. In particular we show that there are provable implications from the $ω^ω$ variants to the $2^ω$ variants and that some of these principles imply all the cardinals in the Cichoń's diagram are large. We also show, similar to (but not the same as) $\mathsf{BA}$, that none of the Lipschitz variants follow from a large fragment of $\mathsf{MA}$.

Figures (1)

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Theorems & Definitions (72)

• Definition 1.1
• Theorem 1.2: See Theorem \ref{['consistency']} below
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Corollary 1.6
• Theorem 1.7
• Proposition 1.8: See Stepranswatson87
• Definition 1.9
• Definition 2.1