Half-flips are 5-avoidable

TL;DR

This paper constructs a 5-letter morphic word avoiding half-flips and establishes bounds on avoidability for half-flips of various lengths, advancing understanding of pattern avoidance in words.

Contribution

It presents a 5-letter morphic word avoiding half-flips and determines avoidability thresholds for different half-flip lengths.

Findings

Constructed a 5-letter morphic word avoiding half-flips.

Half-flips with |u|≥2 are 3-avoidable.

Half-flips with |u|≥4 are 2-avoidable.

Abstract

A word contains a \emph{half-flip} if it contains non-empty factors $uv$ and $vu$ where $|u|=|v|$. Fici reports a non-constructive proof of the existence of an infinite word over a finite alphabet avoiding half-flips and asks for the size of the smallest alphabet over which half-flips may be avoided. Currie and Rampersad have proposed a pure morphic word over 8 letters and a morphic word over 5 letters and conjecture that they avoid half-flips. We present a pure morphic word over 5 letters that avoids half-flips. We also show that half-flips with $|u|\ge2$ are 3-avoidable and that half-flips with $|u|\ge4$ are 2-avoidable.