We prove that the following statement follows from the Open Colouring Axiom (OCA): if X is locally compact σ-compact but not compact and if itš Cech-Stone remainder X * is a continuous image of ω * then X is the union of ω and a compact set. It follows that the remainders of familiar spaces like the real line or the sum of countably many Cantor sets need not be continuous images of ω * .

An n x m rational matrix A is said to be partition regular if for every finite colouring oi \ there is a monochromatic vector x s N with Ax = 0. A set D c f̂J is said to be partition regular for A (or for the system of equations Ax = 0) if for every finite colouring of D there is a monochromatic x e D' with Ax = 6. In this paper we show that for every n there is a set that is partition regular f...

The aim of this article is to obtain a characterization of the canonical extension of Boolean homomorphisms via the Stone-Čech compactification.