Crucial words and the complexity of some extremal problems for sets of prohibited words

We introduce the notion of a set of prohibitions and give definitions of a complete set and a crucial word with respect to a given set of prohibitions. We consider three special sets which appear in different areas of mathematics and for each of them examine the length of a crucial word. One of these sets is proved to be incomplete. The problem of determining lengths of words that are free from a set of prohibitions is shown to be NP-complete, although the related problem of whether or not a given set of prohibitions is complete is known to be effectively solvable.