Square-Free Partial Words with Many Wildcards

We contribute to the study of square-free words. The classical notion of a square-free word has a natural generalization to partial words, studied in several papers since 2008. We prove that the maximal density of wildcards in the ternary infinite square-free partial word is surprisingly big: 3/16. Further we show that the density of wildcards in a finitary infinite square-free partial words is at most 1/3 and this bound is reached by a quaternary word. We demonstrate that partial square-free words can be viewed as "usual" squarefree words with some letters replaced by wildcards and introduce the corresponding characteristic of infinite square-free words, called flexibility. The flexibility is estimated for some important words and classes of words; an interesting phenomenon is the existence of "rigid" square-free words, having no room for wildcards at all.