Half-flips are 5-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.