In Cube-free infinite binary words it was established that there are infinitely many cube-free infinite binary words (see the earlier question for definitions of terms). The construction given in answer to that question yields a countable infinity of such words. In a comment on that answer, I raised the question of whether there is an uncountable infinity of such words. My comment has not generated any response; perhaps it will attract more interest as a question.
I should admit that I ask out of idle curiosity, and have no research interest in the answer; it just seems like the logical question to ask once you know some set is infinite.