How far away must forced letters be so that squares are still avoidable?
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We describe a new non-constructive technique to show that squares are avoidable by an infinite word even if we force some letters from the alphabet to appear at certain occurrences. We show that as long as forced positions are at distance at least 3 (resp. 19, resp. 2) from each other then we can avoid squares over 4 letters (resp. 3 letters, resp. 6 letters). We can also deduce exponential lower bounds on the number of solutions. For our main Theorem to be applicable, we need to check the existence of some languages and we explain how to verify that they exist with a computer. The main purpose of this article is not so much the proofs of these results, but to develop and advertise the method that we use. We hope that this technique could be applied to other avoidability questions where the good approach seems to be non-constructive (e.g., the Thue-list coloring number of the infinite path).
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