Avoiding Multiple Repetitions in Euclidean Spaces

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DĘBSKI Michał Karol GRYTCZUK Jarosław NAYAR Barbara PASTWA Urszula SOKÓŁ Joanna TUCZYŃSKI Michał WENUS Przemysław WE?SEK Krzysztof

Year of publication 2020
Type Article in Periodical
Magazine / Source SIAM Journal on Discrete Mathematics
MU Faculty or unit

Faculty of Informatics

Web https://doi.org/10.1137/18M1180347
Doi http://dx.doi.org/10.1137/18M1180347
Keywords nonrepetitive sequence; pattern avoidance; Euclidean Ramsey theory; Lovász Local Lemma
Description We study colorings of Euclidean spaces avoiding specified patterns on straight lines. This extends the seminal work of Thue on avoidability properties of sequences to continuous, higher dimensional structures. We prove that every space R^d has a 2-coloring such that no sequence of colors derived from collinear points separated by unit distance consists of more than r(d) identical blocks. In case of the plane we show that r(2) <= 43. We also consider more general patterns and give a sufficient condition for a pattern to be avoided in the plane. This supports a general Pattern Avoidance Conjecture in Euclidean spaces. The proofs are based mainly on the probabilistic method, but additional tools are forced by the geometric nature of the problem. We also consider similar questions for general geometric graphs in the plane. In the conclusion of the paper, we pose several conjectures alluding to some famous open problems in Euclidean Ramsey Theory.
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