A Ramsey characterisation of eventually periodic words.

A factorisation x=u1u2⋯$x = u_1 u_2 \cdots$ 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 ui$u_i$ 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=u1u2⋯$x = u_1 u_2 \cdots$ is called 'super‐monochromatic' if each word uk1uk2⋯ukn$u_{k_1} u_{k_2} \cdots u_{k_n}$, where k1<⋯<kn$k_1 < \cdots < k_n$, 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. [ABSTRACT FROM AUTHOR]