Combinatorial Properties Related to the Higher Baumgartner's Axiom

TL;DR

This paper identifies two combinatorial properties related to the higher Baumgartner's axiom, showing their consistency with CH and how their conjunction influences forcing extensions to realize the axiom.

Contribution

It introduces two specific combinatorial properties expressible by a $ ext{Pi}_2$-sentence and explores their implications for forcing and the higher Baumgartner's axiom.

Findings

Each property is consistent with CH.

Their conjunction with certain cardinal equalities implies a c.c.c. forcing to obtain the higher Baumgartner's axiom.

The properties are expressible in a specific structure involving $H(oldsymbol{ ext{omega}}_3)$.

Abstract

We isolate two combinatorial properties, each expressible by a $\Pi_2$-sentence over the structure $(H(\omega_3),\in,\omega_1,\omega_2,\text{NS}_{\omega_2})$, such that each property is consistent with CH, and their conjunction together with $2^\omega \le \omega_2$ and $2^{\omega_1} = 2^{\omega_2} = \omega_3$ implies the existence of a c.c.c. forcing which forces the higher Baumgartner's axiom.