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Non-spherical sets versus lines in Euclidean Ramsey theory

Part of: Discrete geometry Extremal combinatorics

Published online by Cambridge University Press:  22 August 2025

David Conlon  and
Jakob Führer Open the ORCID record for Jakob Führer [Opens in a new window]
David Conlon*
Affiliation:
Department of Mathematics, https://ror.org/05dxps055 California Institute of Technology , Pasadena, CA 91125, United States
Jakob Führer
Affiliation:
Institute for Algebra, https://ror.org/052r2xn60 Johannes Kepler University , 4040 Linz, Austria e-mail: jakob.fuehrer@jku.at
*
e-mail: dconlon@caltech.edu
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Abstract

We show that for every non-spherical set X in $\mathbb {E}^d$, there exists a natural number m and a red/blue-coloring of $\mathbb {E}^n$ for every n such that there is no red copy of X and no blue progression of length m with each consecutive point at distance $1$. This verifies a conjecture of Wu and the first author.

Keywords

Euclidean Ramsey theoryspherical setsequidistribution

MSC classification

Primary: 05D10: Ramsey theory
Secondary: 52C10: Erdős problems and related topics of discrete geometry

Information

Type
Article
Information
Canadian Mathematical Bulletin , First View , pp. 1 - 5
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Canadian Mathematical Society

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Footnotes

D.C. was supported by NSF Awards DMS-2054452 and DMS-2348859. J.F. was supported by the Austrian Science Fund (FWF) under the project W1230.

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