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Could it be that the above questions have resisted solution because we haven’t tapped into the full strength of “proper?” Let us focus on the second question for a moment. Consider the following analogy. Recall that a forcing Q satisfies the countable chain condition (c.c.c.) if every uncountable collection of conditions in Q contains two compatible… (More)

In this paper I will show that it is relatively consistent with the usual axioms of mathematics (ZFC) together with a strong form of the axiom of infinity (the existence of a supercompact cardinal) that the class of uncountable linear orders has a five element basis. In fact such a basis follows from the Proper Forcing Axiom, a strong form of the Baire… (More)

In [23], Todorcevic gives a survey of basis problems in combinatorial set theory, listing nine theorems and six working conjectures — all in the presence of PFA — including the following three of interest to us here: Conjecture 1 (Todorcevic; [23]). If R is a binary relation, then either R ≤ א0 ·ω1 or [ω1]0 ≤ R. Conjecture 2 (Hajnal, Juhasz; [5]). If X is a… (More)

The purpose of this note is to prove a lower bound on the growth of Følner functions for Thompson’s group F . Specifically I will prove that, for any finite generating set Γ ⊆ F , there is a constant C such that FølF,Γ(C) ≥ expn(0).

In this paper we will characterize — under appropriate axiomatic assumptions — when a linear order is minimal with respect to not being a countable union of scattered suborders. We show that, assuming PFA, the only linear orders which are minimal with respect to not being σ-scattered are either Countryman types or real types. We also outline a plausible… (More)

In [13] it was demonstrated that the Proper Forcing Axiom implies that there is a five element basis for the class of uncountable linear orders. The assumptions needed in the proof have consistency strength of at least infinitely many Woodin cardinals. In this paper we reduce the upper bound on the consistency strength of such a basis to something less than… (More)

The purpose of this article is to connect the notion of the amenability of a discrete group with a new form of structural Ramsey theory. The Ramsey theoretic reformulation of amenability constitutes a considerable weakening of the Følner criterion. As a by-product, it will be shown that in any non amenable group G, there is a subset E of G such that no… (More)

Remark. The notes which follow reflect the content of a two day tutorial which took place at the Fields Institute on 5/29 and 5/30 in 2009. Most of the content has existed in the literature for some time (primarily in the original edition of [10]) but has proved difficult to read and digest for various reasons. The only new material contained in these… (More)

One way to formulate the Baire Category Theorem is that no compact space can be covered by countably many nowhere dense sets. Soon after Cohen’s discovery of forcing, it was realized that it was natural to consider strengthenings of this statement in which one replaces countably many with א1-many. Even taking the compact space to be the unit interval, this… (More)

The purpose of this article is to prove that the forcing axiom for completely proper forcings is inconsistent with the Continuum Hypothesis. This answers a longstanding problem of Shelah.