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We study families of infinite block sequences of elements of the space$\FIN_k$. In particular we study Ramsey properties of such families and Ramseyproperties localized to a selective or semiselective coideal. We show how thestable ordered-union ultrafilters defined by Blass, and Matet-adequate familiesdefined by Eisworth in the case $k=1$ fit in the theory of the Ramsey space ofinfinite block sequences of finite sets of natural numbers.

Ramsey subsets of the space of infinite block sequences of vectors