Parametrizing the abstract Ellentuck theorem

We give a parametrization with perfect subsets of 2∞2∞ of the abstract Ellentuck theorem (see [T.J. Carlson, S.G. Simpson, Topological Ramsey Theory, in: J. Neŝetrˆil, V. Rödl (Eds.), Mathematics of Ramsey Theory, Algorithms and Combinatorics, vol. 5, Springer, Berlin, 1990, pp. 172–183], [S. Todorcevic, Introduction to Ramsey spaces, to appear] or [S. Todorcevic, Lecture notes from a course given at the Fields Institute in Toronto, Canada, Autumn 2002]). The main tool for achieving this goal is a sort of parametrization of an abstract version of the Nash–Williams theorem. As corollaries, we obtain some known classical results like the parametrized version of the Galvin–Prikry theorem due to Miller and Todorcevic [A.W. Miller, Infinite combinatorics and definability, Ann. Pure Appl. Logic 41 (1989) 179–203], and the parametrized version of Ellentuck's theorem due to Pawlikowski [Parametrized Ellentuck theorem, Topology Appl. 37 (1990) 65–73]. Also, we obtain parametized vesions of nonclassical results such as Milliken's theorem [K.R. Milliken, Ramsey's theorem with sums or unions, J. Combin. Theory (A) 18 (1975) 276–290], and we prove that the family of perfectly Ramsey subsets of 2∞×FINk[∞] is closed under the Souslin operation.