Infinite monochromatic sumsets for colourings of the reals
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- by Péter Komjáth, Imre Leader, Paul A. Russell, Saharon Shelah, Dániel T. Soukup and Zoltán Vidnyánszky
- Proc. Amer. Math. Soc. 147 (2019), 2673-2684
- 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.References
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Bibliographic Information
- Péter Komjáth
- Affiliation: Institute of Mathematics, Eötvös University Budapest, Pázmány P. s. 1/C 1117, Budapest, Hungary
- MR Author ID: 104465
- Email: kope@cs.elte.hu
- Imre Leader
- Affiliation: Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WB, United Kingdom
- MR Author ID: 111480
- Email: i.Leader@dpmms.cam.ac.uk
- Paul A. Russell
- Affiliation: Churchill College, University of Cambridge, Cambridge CB3 0DS, United Kingdom
- MR Author ID: 772196
- 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
- MR Author ID: 160185
- ORCID: 0000-0003-0462-3152
- 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
- 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
- © Copyright 2019 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 147 (2019), 2673-2684
- MSC (2010): Primary 03E02, 03E35, 05D10
- DOI: https://doi.org/10.1090/proc/14431
- MathSciNet review: 3951442