6 - Kingman, category and combinatorics

Summary

Abstract

Kingman's Theorem on skeleton limits—passing from limits as n → ∞ along nh (n ∈ ℕ) for enough h > 0 to limits as t → ∞ for t ∈ ℝ— is generalized to a Baire/measurable setting via a topological approach. We explore its affinity with a combinatorial theorem due to Kestelman and to Borwein and Ditor, and another due to Bergelson, Hindman and Weiss. As applications, a theory of ‘rational’ skeletons akin to Kingman's integer skeletons, and more appropriate to a measurable setting, is developed, and two combinatorial results in the spirit of van der Waerden's celebrated theorem on arithmetic progressions are given.

Keywords Baire property, bitopology, complete metrizability, density topology, discrete skeleton, essential contiguity, generic property, infinite combinatorics, measurable function, Ramsey theory, Steinhaus theory

AMS subject classification (MSC2010) 26A03

Introduction

The background to the theme of the title is Feller's theory of recurrent events. This goes back to Feller in 1949 [F1], and received its first textbook synthesis in [F2] (see e.g. [GS] for a recent treatment). One is interested in something (‘it’, let us say for now—we can proceed informally here, referring to the above for details) that happens (by default, or by fiat) at time 0, may or may not happen at discrete times n = 1, 2, …, and is such that its happening ‘resets the clock’, so that if one treats this random time as a new time-origin, the subsequent history is a probabilistic replica of the original situation.