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

We consider several variants of Baumgartner’s axiom for ${\mathrm{\aleph }}_1$-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 $\mathsf{B}\mathsf{A}$, 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 ${\omega }^{\omega }$ variants to the $2^{\omega }$ 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) $\mathsf{B}\mathsf{A}$, that none of the Lipschitz variants follow from a large fragment of $\mathsf{M}\mathsf{A}$.