This paper establishes a refinement of the classical LÃ¶wenheimSkolem theorem. The main result shows that any first order structure has a countable elementary substructure with strong second orderâ€¦ (More)

Motivated by Tukey classification problems and building on work in Part 1 [5], we develop a new hierarchy of topological Ramsey spaces RÎ±, Î± < Ï‰1. These spaces form a natural hierarchy of complexity,â€¦ (More)

We show that 1, w, ax, u x ux and ["iF" are the only cofinal types of directed sets of size S,, but that there exist many cofinal types of directed sets of size continuum. A partially ordered set Dâ€¦ (More)

Motivated by a Tukey classification problem we develop here a new topological Ramsey space R1 that in its complexity comes immediately after the classical Ellentuck space [8]. Associated with R1 isâ€¦ (More)

We study Tukey types of ultrafilters on Ï‰, focusing on the question of when Tukey reducibility is equivalent to Rudin-Keisler reducibility. We give several conditions under which this equivalenceâ€¦ (More)

In [23], Todorcevic gives a survey of basis problems in combinatorial set theory, listing nine theorems and six working conjectures â€” all in the presence of PFA â€” including the following three ofâ€¦ (More)

We show how to force two strong positive partition relations on u, and use them in considering several well-known open problems. In [32] SierpiÃ±ski proved that the well-known Ramsey Theorem [27] doesâ€¦ (More)

We introduce and study a metric notion for trees and relate it to a conjecture of Shelah [10] about the existence of a finite basis for a class of linear orderings.