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Pairwise sums in colourings of the reals

Pairwise sums in colourings of the reals Suppose that we have a finite colouring of $$\mathbb R$$ R . What sumset-type structures can we hope to find in some colour class? One of our aims is to show that there is such a colouring for which no uncountable set has all of its pairwise sums monochromatic. We also show that there is such a colouring such that there is no infinite set X with $$X+X$$ X + X (the pairwise sums from X, allowing repetition) monochromatic. These results assume CH. In the other direction, we show that if each colour class is measurable, or each colour class is Baire, then there is an infinite set X (and even an uncountable X, of size the reals) with $$X+X$$ X + X monochromatic. We also give versions for all of these results for k-wise sums in place of pairwise sums. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg Springer Journals

Pairwise sums in colourings of the reals

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References (10)

Publisher
Springer Journals
Copyright
Copyright © 2016 by The Author(s)
Subject
Mathematics; Mathematics, general; Algebra; Differential Geometry; Number Theory; Topology; Geometry
ISSN
0025-5858
eISSN
1865-8784
DOI
10.1007/s12188-016-0166-x
Publisher site
See Article on Publisher Site

Abstract

Suppose that we have a finite colouring of $$\mathbb R$$ R . What sumset-type structures can we hope to find in some colour class? One of our aims is to show that there is such a colouring for which no uncountable set has all of its pairwise sums monochromatic. We also show that there is such a colouring such that there is no infinite set X with $$X+X$$ X + X (the pairwise sums from X, allowing repetition) monochromatic. These results assume CH. In the other direction, we show that if each colour class is measurable, or each colour class is Baire, then there is an infinite set X (and even an uncountable X, of size the reals) with $$X+X$$ X + X monochromatic. We also give versions for all of these results for k-wise sums in place of pairwise sums.

Journal

Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg – Springer Journals

Published: Dec 21, 2016

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