Half-flips are 5-avoidable

2026-05-12

AI Summary

This study investigates the minimal alphabet size required to avoid “half-squares”—factors of the form $uv$ and $vu$ with $u$ and $v$ nonempty and of equal length—in infinite words. By constructing purely morphic words and employing combinatorial word-theoretic analysis, the authors present the first purely morphic word over a five-letter alphabet that completely avoids all half-squares. Moreover, they establish tighter bounds under length constraints: when $|u| \geq 2$, half-squares can be avoided over a ternary alphabet, and when $|u| \geq 4$, avoidance is achievable even over a binary alphabet. These results delineate sharp thresholds for half-square avoidability under varying length conditions, thereby advancing the theory of repetition-free sequences in combinatorics on words.

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.

Problem

• half-flips
• avoidability
• combinatorics on words
• alphabet size
• pattern avoidance

Innovation

• half-flip avoidance
• pure morphic word
• avoidability
• combinatorics on words
• alphabet size