On Non-Complete Sets and Restivos Conjecture

A finite set S of words over the alphabet A is called non-complete if Fact(S*) is different from A*. A word w in A* - Fact(S*) is said to be uncompletable. We present a series of non-complete sets S_k whose minimal uncompletable words have length 5k^2 - 17k + 13, where k > 3 is the maximal length of words in S_k. This is an infinite series of counterexamples to Restivo’s conjecture, which states that any non-complete set possesses an uncompletable word of length at most 2k^2.