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 \({\scriptstyle \sum }\) -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.
