Gaps in $(\mathcal {P}(\omega ),\subset ^*)$ and $(\omega ^{\omega },\le ^*)$

Abstract:

For a partial order $(P,\le _P)$, let $\Gamma (P,\le _P)$ denote the statement that for every $\le _P$-increasing $\omega _1$-sequence $a\subseteq P$ there is a $\le _P$-decreasing $\omega _1$-sequence $b\subseteq P$ on top of $a$ such that $(a,b)$ is an $(\omega _1,\omega _1)$-gap in $P$. The main result of this paper is that $\mathfrak t>\omega _1\leftrightarrow \Gamma (\mathcal P(\omega ),\subset ^*)\leftrightarrow \Gamma (\omega ^\omega ,\le ^*)$. It is also shown, as a corollary, that $\Gamma (\omega ^\omega ,\le ^*)\to \mathfrak b>\omega _1$ but $\mathfrak b>\omega _1\not \to \Gamma (\omega ^\omega ,\le ^*)$.