Avoiding Multiple Repetitions in Euclidean Spaces
Michał Dębski, Jarosław Grytczuk, Barbara Nayar, Urszula Pastwa, Joanna Sokół, Michał Tuczyński, Przemysław Wenus, Krzysztof Węsek · SIAM Journal on Discrete Mathematics · 2020
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 $\mathbb{R}^d$ 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 $r(2)\leqslant 43$. 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.