Non-repetitive colorings of infinite sets

In this paper we investigate colorings of sets avoiding similarly colored subsets. If S is an arbitrary colored set and T is a fixed family of bijections of S to itself, then two subsets A,B⊆S are said to be colored similarly with respect to T, if there exists a transformation t∈T mapping A onto B, and preserving a coloring of A. This general setting covers some well-known topics such as non-repetitive sequences of Thue or the famous Hadwiger–Nelson problem on unit distances in Euclidean spaces. Our main theorem of this paper concerns arbitrary infinite sets, however, the most striking consequences are obtained for the case of Euclidean spaces. For instance, there exist 2-colorings of Rn with no two different line segments colored similarly, with respect to translations. The method is based on the principle of induction, hence it is not constructive in general, and the problem of explicit constructions arises naturally. We give two such examples of non-repetitive colorings of the sets R and Q, with respect to translations. In conclusion of the paper we discuss possible generalizations and pose two open problems.