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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Poset limits can be totally ordered


Authors: Jan Hladký, András Máthé, Viresh Patel and Oleg Pikhurko
Journal: Trans. Amer. Math. Soc. 367 (2015), 4319-4337
MSC (2010): Primary 06A06, 28Axx
DOI: https://doi.org/10.1090/S0002-9947-2015-06299-0
Published electronically: February 3, 2015
MathSciNet review: 3324929
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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 Szemerédi-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.


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Additional Information

Jan Hladký
Affiliation: Mathematics Institute, University of Warwick, Coventry, CV4 7AL, United Kingdom
Address at time of publication: Institute of Mathematics, Academy of Sciences of the Czech Republic, Žitná 25, 110 00, Praha, Czech Republic
Email: J.Hladky@warwick.ac.uk, honzahladky@gmail.com

András Máthé
Affiliation: Mathematics Institute, University of Warwick, Coventry, CV4 7AL, United Kingdom
Email: A.Mathe@warwick.ac.uk

Viresh Patel
Affiliation: School of Mathematics, Birmingham University, Edgbaston, Birmingham B15 2TT, United Kingdom
Address at time of publication: School of Mathematical Sciences, Queen Mary, University of London, Mile End Road, London E1 4NS, United Kingdom
Email: viresh.s.patel@googlemail.com

Oleg Pikhurko
Affiliation: Mathematics Institute and DIMAP, University of Warwick, Coventry, CV4 7AL, United Kingdom

DOI: https://doi.org/10.1090/S0002-9947-2015-06299-0
Keywords: Homomorphism density, ordered probability space, partially ordered set, poset kernel, Regularity Lemma
Received by editor(s): November 11, 2012
Received by editor(s) in revised form: August 2, 2013
Published electronically: February 3, 2015
Additional Notes: The first author was supported by an EPSRC fellowship.
The second author was supported by the EPSRC (grant EP/G050678/1), the Hungarian Scientific Research Fund (grants 72655 and 104178) and the Leverhulme Trust.
The third author was supported by the EPSRC (grant EP/J008087/1).
The fourth author was supported by the European Research Council (grant agreement no. 306493) and the National Science Foundation of the USA (grant DMS-1100215).
Article copyright: © Copyright 2015 American Mathematical Society

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