Forced repetitions over alphabet lists

Thue [14] showed that there exist arbitrarily long square-free strings over an alphabet of three symbols (not true for two symbols). An open problem was posed in [7], which is a generalization of Thue's original result: Given an alphabet list L = L1, . . . ,Ln, where |Li| = 3, is it always possible to find a square-free string, w = w1w2 wn, where wi ∈ Li? In this paper we show that squares can be forced on square-free strings over alphabet lists iff a suffix of the square-free string conforms to a pattern which we term as an offending suffix. We also prove properties of offending suffixes. However, the problem remains tantalizingly open.