Pattern Avoiding Colorings of Euclidean Spaces

Abstract

Given a coloring \(f\) of Euclidean space \(\mathbb{R}^n\) and some group \(G\) of its transformations, its subsets \(A\) and \(B\) are said to be colored similarly, if there exists \(g \in G\), such that \(B = g(A)\) and \(f(a) = f(g(a))\), for all \(a \in A\). From our earlier result [12] it follows that there are \(2\)-colorings of \(\mathbb{R}^n\), in which no two different line segments are colored similarly with respect to isometries. The main purpose of this paper is to investigate other types of such pattern avoiding colorings. In particular, we consider topological as well as measure theoretic aspects of the above scene. Our motivation for studying this topic is twofold. One is that it extends square-free colorings of \(\mathbb{R}\), introduced in [2] as a continuous version of the famous non-repetitive sequences of Thue. The other is its relationship to some exciting problems and results of Euclidean Ramsey Theory, especially those concerning avoiding distances.