A parametrization of the abstract Ramsey theorem

We give a parametrization with perfect subsets of 2∞ of the abstract Ramsey theorem (see [13]). Our main tool is an adaptation, to a more general context of Ramsey spaces, of the techniques developed in [8] by J. G. Mijares in order to obtain the corresponding result within the context of topological Ramsey spaces. This tool is inspired by Todorcevic’s abstract version of the combinatorial forcing introduced by Galvin and Prikry in [6], and also by the parametrized version of this combinatorial technique, developed in [12] by Pawlikowski. The main result obtained in this paper (theorem 5 below) turns out to be a generalization of the parametrized Ellentuck theorem of [8], and it yields as corollary that the family of perfectly Ramsey sets corresponding to a given Ramsey space is closed under the Souslin operation. This enabled us to prove a parametrized version of the infinite dimensional Hales-Jewett theorem (see [13]).