WM groups and Ramsey theory

Wm Groups and Ramsey Theory

Our goal in this paper is to exhibit a connection between two seemingly disparate areas: Ramsey theory and the theory of unitary representations of a class of locally compact groups. The class of groups that we are interested in consists of the so-called ”minimally periodic groups” introduced by von Neumann in [N]; namely, groups having the property that they do not admit non-trivial almost per...

Amenability and Ramsey Theory

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 finitely additive probability measure on G...

Ramsey Theory

Ramsey Theory

Ramsey Theory is the study of inevitable substructures in large (usually discrete) objects. For example, consider colouring the edges of the complete graph Kn with two colours. In 1930, Ramsey [13] proved that if n is large enough, then we can find either a red complete subgraph on k vertices or a blue complete subgraph on ` vertices. We write Rpk, `q for the smallest such n. Another famous exa...

Ramsey Theory

We give a proof to arithmetic Ramsey’s Theorem. In addition, we show the proofs for Schur’s Theorem, the Hales-Jewett Theorem, Van der Waerden’s Theorem and Rado’s Theorem, which are all extensions of the classical Ramsey’s Theorem.