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Infinite monochromatic sumsets for colourings of the reals


Authors: Péter Komjáth, Imre Leader, Paul A. Russell, Saharon Shelah, Dániel T. Soukup and Zoltán Vidnyánszky
Journal: Proc. Amer. Math. Soc. 147 (2019), 2673-2684
MSC (2010): Primary 03E02, 03E35, 05D10
DOI: https://doi.org/10.1090/proc/14431
Published electronically: March 5, 2019
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Abstract: N. Hindman, I. Leader, and D. Strauss proved that it is consistent that there is a finite colouring of $ \mathbb{R}$ so that no infinite sumset $ X+X$ is monochromatic. Our aim in this paper is to prove a consistency result in the opposite direction: we show that, under certain set-theoretic assumptions, for any finite colouring $ c$ of $ \mathbb{R}$ there is an infinite $ X\subseteq \mathbb{R}$ so that $ c\upharpoonright X+X$ is constant.


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Additional Information

Péter Komjáth
Affiliation: Institute of Mathematics, Eötvös University Budapest, Pázmány P. s. 1/C 1117, Budapest, Hungary
Email: kope@cs.elte.hu

Imre Leader
Affiliation: Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WB, United Kingdom
Email: i.Leader@dpmms.cam.ac.uk

Paul A. Russell
Affiliation: Churchill College, University of Cambridge, Cambridge CB3 0DS, United Kingdom
Email: p.a.russell@dpmms.cam.ac.uk

Saharon Shelah
Affiliation: Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Jerusalem, 91904, Israel – and – Department of Mathematics, Rutgers University, New Brunswick, New Jersey 08854
Email: shelah@math.huji.ac.il

Dániel T. Soukup
Affiliation: Universität Wien, Kurt Gödel Research Center for Mathematical Logic, 1090 Wien, Austria
Email: daniel.soukup@univie.ac.at

Zoltán Vidnyánszky
Affiliation: Universität Wien, Kurt Gödel Research Center for Mathematical Logic, 1090 Wien, Austria – and – Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Hungary
Email: vidnyanszkyz@gmail.com

DOI: https://doi.org/10.1090/proc/14431
Keywords: Sumset, monochromatic, colouring, partition relation, continuum
Received by editor(s): December 29, 2017
Received by editor(s) in revised form: October 2, 2018
Published electronically: March 5, 2019
Additional Notes: The fifth author was supported in part by PIMS, the National Research, Development and Innovation Office–NKFIH grant no. 113047 and the FWF Grant I1921. This research was partially done whilst visiting the Isaac Newton Institute for Mathematical Sciences part of the programme ‘Mathematical, Foundational and Computational Aspects of the Higher Infinite’ (HIF) funded by EPSRC grant EP/K032208/1. The fifth author is the corresponding author.
The sixth author was partially supported by the National Research, Development and Innovation Office–NKFIH grants no. 113047, no. 104178, and no. 124749 and by FWF Grant P29999.
Communicated by: Heike Mildenberger
Article copyright: © Copyright 2019 American Mathematical Society