Pairwise sums in colourings of the reals

Abstract

Suppose that we have a finite colouring of the reals. 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 (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 monochromatic. We also give versions for all of these results for k-wise sums in place of pairwise sums.

References in corpus (27)

• PYTHIA 6.4 Physics and Manual
• The spectrum of isotropic diffuse gamma-ray emission between 100 MeV and 820 GeV
• A combined maximum-likelihood analysis of the high-energy astrophysical neutrino flux measured with IceCube
• Dark matter production in the early Universe: beyond the thermal WIMP paradigm
• Flavor Ratio of Astrophysical Neutrinos above 35 TeV in IceCube
• Coherent Propagation of PeV Neutrinos and the Dip in the Neutrino Spectrum at IceCube
• Theoretically palatable flavor combinations of astrophysical neutrinos
• Testing the Dark Matter Scenario for PeV Neutrinos Observed in IceCube
• Spectral analysis of the high-energy IceCube neutrinos
• IceCube events and decaying dark matter: hints and constraints
• Galactic Neutrinos in the TeV to PeV Range
• Boosted Dark Matter in IceCube and at the Galactic Center
• Neutrinoful Universe
• Mind the gap on Icecube: Cosmic neutrino spectrum and muon anomalous magnetic moment in the gauged L_μ - L_τ model
• Decaying Leptophilic Dark Matter at IceCube
• The Direct Detection of Boosted Dark Matter at High Energies and PeV events at IceCube
• New Search for Monochromatic Neutrinos from Dark Matter Decay
• Possible Interpretations of IceCube High-Energy Neutrino Events
• Galactic PeV neutrinos from dark matter annihilation
• PeV Neutrinos and a 3.5 keV X-Ray Line from a PeV Scale Supersymmetric Neutrino Sector
• Neutrino Portal Dark Matter: From Dwarf Galaxies to IceCube
• Diffuse neutrinos from extragalactic supernova remnants: Dominating the 100 TeV IceCube flux
• Star-forming galaxies as the origin of the IceCube PeV neutrinos
• Implications of Fermi-LAT observations on the origin of IceCube neutrinos
• Decaying Dark Matter in the Supersymmetric Standard Model with Freeze-in and Seesaw mechanims
• AMS02 positron excess from decaying fermion DM with local dark gauge symmetry
• Neutrino properties from ultra-high energy cosmic neutrinos

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