Partitioning pairs of countable ordinals (Q1111550)

It is proved that \(\omega_ 1\nrightarrow [\omega_ 1]^ 2_{\omega_ 1}\). The paper not only contains this striking, deep, new result, but a careful exposition of the completely new method applied here, the method of ``Todorcevic walks''. Among the applications given one can find short proofs for the existence of Countryman types, Souslin trees (when a Cohen real is added). The results are also extended to larger cardinals, for those with a nonreflecting stationary set for the original result, and for those having a square sequence for other results. A powerful stepping up lemma is proved. It is sketched how other extensions are given (mostly by S. Shelah).