Nonrepetitive colorings of line arrangements

Abstract

A sequence $S$ is nonrepetitive if no two adjacent segments of $S$ are identical. A famous result of Thue from 1906 asserts that there are arbitrarily long nonrepetitive sequences over 3 symbols. We study the following geometric variant of this problem. Given a set $P$ of points in the plane and a set $L$ of lines, what is the least number of colors needed to color $P$ so that every line in $L$ is nonrepetitive? If $P$ consists of all intersection points of a prescribed set of lines $L$, then we prove that there is such coloring using at most 405 colors. The proof is based on a theorem of Thue and on a result of Alon and Marshall concerning homomorphisms of edge colored planar graphs. We also consider nonrepetitive colorings involving other geometric structures. For instance, a nonrepetitive analog of the famous Hadwiger–Nelson problem is formulated as follows: what is the least number of colors needed to color the plane so that every path of the unit distance graph whose vertices are colinear is nonrepetitive? Using a theorem of Thue we prove that this number is at most $36$.