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

In 2007, Grytczuk conjecture that for any sequence (li)i≥1 of alphabets of size 3 there exists a square-free infinite word w such that for all i, the ith letter of w belongs to li. The result of Thue of 1906 implies that there is an infinite square-free word if all the li are identical. On the other, hand Grytczuk, Przyby lo and Zhu showed in 2011 that it also holds if the li 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 (the previous similar bound was 2). 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 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).