Products of Directed Sets with Calibre $(\omega_1, \omega)$

Abstract

A directed set $P$ is calibre $(\omega_1, \omega)$ if every uncountable subset of $P$ contains an infinite bounded subset. $P$ is productively calibre $(\omega_1, \omega)$ if $P \times Q$ is calibre $(\omega_1, \omega)$ for every directed set $Q$ with calibre $(\omega_1, \omega)$, and $P$ is powerfully calibre $(\omega_1, \omega)$ if the countable power of $P$ is calibre $(\omega_1, \omega)$. It is shown that (1) uncountable products are calibre $(\omega_1, \omega)$ only in highly restrictive circumstances, (2) many but not all $\sum$-products of calibre $(\omega_1, \omega)$ directed sets are calibre $(\omega_1, \omega)$, (3) there are directed sets which are calibre $(\omega_1, \omega)$ but neither productively nor powerfully calibre $(\omega_1, \omega)$, and (4) there are directed sets which are powerfully but not productively calibre $(\omega_1, \omega)$. As an application, the position is established of $\sum \omega^{\omega_1}$ in the Tukey order among Isbell's classical 10 directed sets.