The strength of infinitary Ramseyan principles can be accessed by their densities

Abstract

In this article, we conduct a model-theoretic investigation of three infinitary Ramseyan statements: the Infinite Ramsey Theorem for pairs and two colours (RT22), the Canonical Ramsey Theorem for pairs (CRT2) and the Regressive Ramsey Theorem for pairs (RegRT2). We approximate the logical strength of these principles by the strength of their first-order iterated versions, known as density principles. We then investigate their logical strength using strong initial segments of models of Peano Arithmetic, in the spirit of the classical article by Paris and Kirby, hereby re-proving old results model-theoretically. The article is concluded by a discussion of two further outreaches of densities. One is a further investigation of the strength of the Ramsey Theorem for pairs. The other deals with the asymptotics of the standard Ramsey function R22.