Avoiding squares over words with lists of size three amongst four symbols

Abstract:

In 2007, Grytczuk conjectured that for any sequence $(\ell _i)_{i\ge 1}$ of alphabets of size $3$ there exists a square-free infinite word $w$ such that for all $i$, the $i$-th letter of $w$ belongs to $\ell _i$. The result of Thue from 1906 implies that there is an infinite square-free word if all the $\ell _i$ are identical. On the other hand, Grytczuk, Przybyło and Zhu showed in 2011 that it also holds if the $\ell _i$ are of size $4$ instead of $3$.

In this article, we first show that if the lists are of size $4$, the number of square-free words is at least $2.45^n$ (the previous similar bound was $2^n$). We then show our main result: we can construct such a square-free word if the lists are subsets of size $3$ of the same alphabet of size $4$. Our proof also implies that there are at least $1.25^n$ square-free words of length $n$ for any such list assignment. This proof relies on the existence of a set of coefficients verified with a computer. We suspect that the full conjecture could be resolved by this method with a much more powerful computer (but we might need to wait a few decades for such a computer to be available).