Combinatorial Properties Related to the Higher Baumgartner's Axiom

Abstract

We isolate two combinatorial properties, each expressible by a ${\hat{I}}_2$-sentence over the structure $(H(\ddot{I}{‰}_3),\in ,\ddot{I}{‰}_1,\ddot{I}{‰}_2,{\text{NS}}_{\ddot{I}{‰}_2})$, such that each property is consistent with CH, and their conjunction together with $2^{\ddot{I}}‰\le \ddot{I}{‰}_2$ and $2^{\ddot{I}{‰}_1}=2^{\ddot{I}{‰}_2}=\ddot{I}{‰}_3$ implies the existence of a c.c.c. forcing which forces the higher Baumgartner's axiom.