Block combinatorics

In this paper we extend the block combinatorics partition theorems of Hindman and Milliken-Taylor in the setting of the recursive system of the block Schreier families , consisting of families defined for every countable ordinal . Results contain (a) a block partition Ramsey theorem for every countable ordinal (Hindman's Theorem corresponding to , and the Milliken-Taylor Theorem to a finite ordinal), (b) a countable ordinal form of the block Nash-Williams partition theorem, and (c) a countable ordinal block partition theorem for sets closed in the infinite block analogue of Ellentuck's topology.