Infinite monochromatic sumsets for colourings of the reals

Péter Komjáth; Imre Leader; Paul A. Russell; Saharon Shelah; Dániel T. Soukup; Zoltán Vidnyánszky

doi:10.1090/proc/14431

Infinite monochromatic sumsets for colourings of the reals

Péter Komjáth
, Imre Leader
, Paul A. Russell
, Saharon Shelah
, Dániel T. Soukup
, Zoltán Vidnyánszky

Einstein Institute of Mathematics

Research output: Contribution to journal › Article › peer-review

8 Scopus citations

Abstract

N. Hindman, I. Leader, and D. Strauss proved that it is consistent that there is a finite colouring of 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 R there is an infinite X ⊆ R so that c ↾ X + X is constant.

Original language	English
Pages (from-to)	2673-2684
Number of pages	12
Journal	Proceedings of the American Mathematical Society
Volume	147
Issue number	6
DOIs	https://doi.org/10.1090/proc/14431
State	Published - 2019

Bibliographical note

Publisher Copyright:
© 2019 Amerian Mathematial Soiety.

Keywords

Colouring
Continuum
Monochromatic
Partition relation
Sumset

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10.1090/proc/14431

Cite this

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title = "Infinite monochromatic sumsets for colourings of the reals",

abstract = "N. Hindman, I. Leader, and D. Strauss proved that it is consistent that there is a finite colouring of 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 R there is an infinite X ⊆ R so that c ↾ X + X is constant.",

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AU - Komjáth, Péter

AU - Leader, Imre

AU - Russell, Paul A.

AU - Shelah, Saharon

AU - Soukup, Dániel T.

AU - Vidnyánszky, Zoltán

PY - 2019

Y1 - 2019

N2 - N. Hindman, I. Leader, and D. Strauss proved that it is consistent that there is a finite colouring of 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 R there is an infinite X ⊆ R so that c ↾ X + X is constant.

AB - N. Hindman, I. Leader, and D. Strauss proved that it is consistent that there is a finite colouring of 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 R there is an infinite X ⊆ R so that c ↾ X + X is constant.

KW - Colouring

KW - Continuum

KW - Monochromatic

KW - Partition relation

KW - Sumset

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JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

IS - 6

ER -

Infinite monochromatic sumsets for colourings of the reals

Abstract

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Keywords

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