José G. Mijares

A notion of selective ultrafilter corresponding to topological Ramsey spaces

We introduce the relation of almost-reduction in an arbitrary topological Ramsey space ℛ as a generalization of the relation of almost-inclusion on ℕ[∞]. This leads us to a type of ultrafilter ⊆ ℛ which corresponds to the well-known notion of selective ultrafilter on ℕ. The relationship turns out to be rather exact in the sense that it permits us to lift several well-known facts about selective ultrafilters on ℕ and the Ellentuck space ℕ[∞] to the ultrafilter and the Ramsey space ℛ. For example, we prove that the open coloring axiom holds on L (ℝ)[], extending therefore the result from [3] which gives the same conclusion for the Ramsey space ℕ[∞]. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Local Ramsey theory: an abstract approach

Given a topological Ramsey space

Ideal games and Ramsey sets

(\mathcal R,\leq, r)

Topological Ramsey spaces and metrically Baire sets

, we extend the notion of semiselective coideal to sets

Parametrizing the abstract Ellentuck theorem

\mathcal H\subseteq\mathcal R

On Galvin’s Lemma and Ramsey Spaces

and study conditions for

A Parametrization of the Abstract Ramsey Theorem

\mathcal H

A Ramsey space of infinite polyhedra and the infinite random polyhedron

that will enable us to make the structure

Coideals of block sequences

(\mathcal R,\mathcal H,\leq, r)

Ramsey subsets of the space of infinite block sequences of vectors

a Ramsey space (not necessarily topological) and also study forcing notions related to