On extendibility of unavoidable sets

Abstract A subset X of a free monoid A∗ is said to be unavoidable if all but finitely many words in A∗ contain some word of X as a subword. A. Ehrenfeucht has conjectured that every unavoidable set X is extendible in the sense that there exist x ϵ X and a ϵ A such that (X − {x}) ∪ {xa} is itself unavoidable. This problem remains open, we give some partial solutions and show how to efficiently test unavoidability, extendibility and other properties of X related to the problem.