We prove that the group of homeomorphisms of the circle introduced by the author in joint work with Justin Moore (Groups, Geometry and Dynamics 2015) is of type F∞. This provides the first example of a type F∞ group which is nonamenable and does not contain non abelian free subgroups. To prove our result we provide a certain generalisation of cube complexes, which we refer to as cluster complexes. We also obtain a normal form, or a canonical unique choice of words for the elements of the group.

In this note we will discuss a new reflection principle which follows from the Proper Forcing Axiom. The immediate purpose will be to prove that the bounded form of the Proper Forcing Axiom implies both that 2 ω = ω 2 and that L(P(ω 1 )) satisfies the Axiom of Choice. It will also be demonstrated that this reflection principle implies that (κ) fails for all regular κ > ω 1 .

Abstract. We will present a collection of guessing principles which have a similar relationship to ♦ as cardinal invariants of the continuum have to CH. The purpose is to provide a means for systematically analyzing ♦ and its consequences. It also provides for a unified approach for understanding the status of a number of consequences of CH and ♦ in models such as those of Laver, Miller, and Sacks.

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 Category Theorem. The elements are X, ω 1 , ω * 1 , C, C * where X is any suborder of the reals of cardinality ℵ 1 and C is any Countryman line. This confirms a longstanding conjecture of Shelah.

This paper is dedicated to Stevo Todorcevic for teaching me how to traverse ω 1and for his inspirational [23].