 Currie resolved this in a 2002 paper, showing that all cycles with 18 or more vertices have Thue number 3.Testing whether a coloring has a repetitive path is in NP, so testing whether a coloring is nonrepetitive is in co-NP, and Manin showed that it is co-NP-complete.The problem of finding such a coloring belongs to formula_1 in the polynomial hierarchy, and again Manin showed that it is complete for this level.