A Ramsey characterisation of eventually periodic words

Abstract

A factorisation x = u 1 u 2 ⋯ $x = u_1 u_2 \cdots$ of an infinite word x $x$ on alphabet X $X$ is called ‘monochromatic’, for a given colouring of the finite words X ∗ $X^*$ on alphabet X $X$ , if each u i $u_i$ is the same colour. Wojcik and Zamboni proved that the word x $x$ is periodic if and only if for every finite colouring of X ∗ $X^*$ there is a monochromatic factorisation of x $x$ . On the other hand, it follows from Ramsey's theorem that, for any word x $x$ , for every finite colouring of X ∗ $X^*$ there is a suffix of x $x$ having a monochromatic factorisation. A factorisation x = u 1 u 2 ⋯ $x = u_1 u_2 \cdots$ is called ‘super-monochromatic’ if each word u k 1 u k 2 ⋯ u k n $u_{k_1} u_{k_2} \cdots u_{k_n}$ , where k 1 < ⋯ < k n $k_1 < \cdots < k_n$ , is the same colour. Our aim in this paper is to show that a word x $x$ is eventually periodic if and only if for every finite colouring of X ∗ $X^*$ there is a suffix of x $x$ having a super-monochromatic factorisation. Our main tool is a Ramsey result about alternating sums that may be of independent interest.