Avoiding squares over words with lists of size three amongst four symbols
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In 2007, Grytczuk conjecture that for any sequence (ℓ_i)_i≥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 ℓ_i. The result of Thue of 1906 implies that there is an infinite square-free word if all the ℓ_i are identical. On the other, hand Grytczuk, Przybyło and Zhu showed in 2011 that it also holds if the ℓ_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).
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