Combinatorial Properties Related to the Higher Baumgartner's Axiom

We isolate two combinatorial properties, each expressible by a ${\mathrm{\Pi }}_2$-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.