Poset limits can be totally ordered
Abstract
S.Janson [Poset limits and exchangeable random posets, Combinatorica 31 (2011), 529--563] defined limits of finite posets in parallel to the emerging theory of limits of dense graphs. We prove that each poset limit can be represented as a kernel on the unit interval with the standard order, thus answering an open question of Janson. We provide two proofs: real-analytic and combinatorial. The combinatorial proof is based on a Szemeredi-type Regularity Lemma for posets which may be of independent interest. Also, as a by-product of the analytic proof, we show that every atomless ordered probability space admits a measure-preserving and almost order-preserving map to the unit interval.
Cite
@article{arxiv.1211.2473,
title = {Poset limits can be totally ordered},
author = {Jan Hladky and Andras Mathe and Viresh Patel and Oleg Pikhurko},
journal= {arXiv preprint arXiv:1211.2473},
year = {2013}
}
Comments
Final version accepted by Transactions of Amer Math Soc, 18 pages