Kingman, category and combinatorics

Details Kingman, category and combinatorics Kingman, category and combinatorics

2010-03-24

arxiv.org/abs/1003.4673v1

Mathematics

Classical Analysis and ODEs

34 pages. To appear in Bingham, N. H., and Goldie, C. M. (eds), Probability and Mathematical Genetics: Papers in Honour of Sir

Scientific paper

Kingman's Theorem on skeleton limits---passing from limits as $n\to \infty $ along $nh$ ($n\in \mathbb{N}$) for enough $h>0$ to limits as $t\to \infty $ for $t\in \mathbb{R}$---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.

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