  A complete partition theory is presented for omega-located words (and
omega-words), namely for located words over an infinite alphabet dominated by a
fixed increasing sequence. This theory strengthens in an essential way the
classical Carlson, Furstenberg-Katznelson, and Bergelson-Blass-Hindman
partition theory for words over a finite alphabet. Consequences of this theory
are strong simultaneous extensions of the classical Hindman, Milliken-Taylor
partition theorem, and of a van der Waerden theorem for general semigroups,
extending results of Hindman-Strauss and Beiglbock.
