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

Abstract

In 2007, Grytczuk conjectured that for any sequence ( ℓ i ) i ≥ 1 (\ell _i)_{i\ge 1} of alphabets of size 3 3 there exists a square-free infinite word w w such that for all i i , the i i -th letter of w w belongs to ℓ i \ell _i . The result of Thue from 1906 implies that there is an infinite square-free word if all the ℓ i \ell _i are identical. On the other hand, Grytczuk, Przybyło and Zhu showed in 2011 that it also holds if the ℓ i \ell _i are of size 4 4 instead of 3 3 . In this article, we first show that if the lists are of size 4 4 , the number of square-free words is at least 2.45 n 2.45^n (the previous similar bound was 2 n 2^n ). We then show our main result: we can construct such a square-free word if the lists are subsets of size 3 3 of the same alphabet of size 4 4 . Our proof also implies that there are at least 1.25 n 1.25^n square-free words of length n 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).