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In 2007, Grytczuk conjecture that for any sequence $(\ell_i)_{i\ge1}$ ofalphabets of size $3$ there exists a square-free infinite word $w$ such thatfor all $i$, the $i$-th letter of $w$ belongs to $\ell_i$. The result of Thueof 1906 implies that there is an infinite square-free word if all the $\ell_i$are identical. On the other, hand Grytczuk, Przyby{\l}o and Zhu showed in 2011that 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 numberof 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 wordif the lists are subsets of size $3$ of the same alphabet of size $4$. Ourproof 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 setof coefficients verified with a computer. We suspect that the full conjecturecould be resolved by this method with a much more powerful computer (but wemight need to wait a few decades for such a computer to be available).

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