Is it decidable if a given word is $2$-avoidable? That is, does there exist an algorithm which can determine for a given pattern whether every infinite binary word contains that pattern?
It seems that the more general question whether it is possible to compute the avoidability index of a given word $w$ (the least $k$ such that $w$ is $k$-avoidable) is open according to this Wikipedia page, but I'm not sure whether this case of it is still open.
If it is still open, what is the current state of progress on the question?