# New results on pseudosquare avoidance

## TL;DR

This paper explores the combinatorial properties of binary words with minimal squares and antisquares, classifies possible avoidance patterns, and proves the existence of infinite words avoiding complex morphic patterns.

## Contribution

It provides a complete classification of minimal square and antisquare binary words and establishes the existence of infinite words avoiding all large morphic patterns.

## Key findings

- Classified possible minimal square and antisquare binary words.
- Proved avoidance of $x p(x)$ and $x t(x)$ patterns.
- Established existence of infinite words avoiding all large morphic patterns.

## Abstract

We start by considering binary words containing the minimum possible numbers of squares and antisquares (where an antisquare is a word of the form $x \overline{x}$), and we completely classify which possibilities can occur. We consider avoiding $x p(x)$, where $p$ is any permutation of the underlying alphabet, and $x t(x)$, where $t$ is any transformation of the underlying alphabet. Finally, we prove the existence of an infinite binary word simultaneously avoiding all occurrences of $x h(x)$ for every nonerasing morphism $h$ and all sufficiently large words $x$.