VARIANTS OF BAUMGARTNER’S AXIOM FOR LIPSCHITZ FUNCTIONS ON BAIRE AND CANTOR SPACE

Abstract We consider several variants of Baumgartner’s axiom for ℵ 1 $\aleph _1$ normal first transfinite cardinal 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 BA $\mathsf {BA}$ sans serif upper B upper 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 ω ω $\omega ^\omega $ omega Superscript omega variants to the 2 ω $2^\omega $ 2 Superscript 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) BA $\mathsf {BA}$ sans serif upper B upper A , that none of the Lipschitz variants follow from a large fragment of MA $\mathsf {MA}$ sans serif upper M upper A .