Kingman, category and combinatorics

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.