# How far away must forced letters be so that squares are still avoidable?

## TL;DR

This paper introduces a non-constructive method to determine how far apart forced letters must be in an infinite word to still avoid squares, providing bounds for different alphabet sizes and demonstrating the technique's potential for broader applications.

## Contribution

The paper presents a novel non-constructive approach to avoid squares in infinite words with forced letter positions, establishing new distance bounds for various alphabet sizes.

## Key findings

- Squares are avoidable with forced letters separated by at least 19, 3, or 2 positions depending on alphabet size.
- Exponential lower bounds are established on the number of solutions.
- The technique relies on computer-verified existence of certain languages.

## Abstract

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 19 (resp. 3, resp. 2) from each other then we can avoid squares over 3 letters (resp. 4 letters, resp. 6 or more 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. 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).