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A factorisation $x = u_1 u_2 cdots$ of an infinite word $x$ on alphabet $X$ is called `monochromatic, for a given colouring of the finite words $X^*$ on alphabet $X$, if each $u_i$ is the same colour. Wojcik and Zamboni proved that the word $x$ is periodic if and only if for every finite colouring of $X^*$ there is a monochromatic factorisation of $x$. On the other hand, it follows from Ramseys theorem that, for textit{any} word $x$, for every finite colouring of $X^*$ there is a suffix of $x$ having a monochromatic factorisation.par A factorisation $x = u_1 u_2 cdots$ is called `super-monochromatic if each word $u_{k_1} u_{k_2} cdots u_{k_n}$, where $k_1 < cdots < k_n$, is the same colour. Our aim in this paper is to show that a word $x$ is eventually periodic if and only if for every finite colouring of $X^*$ there is a suffix of $x$ having a super-monochromatic factorisation. Our main tool is a Ramsey result about alternating sums that may be of independent interest.
We define a new class of shift spaces which contains a number of classes of interest, like Sturmian shifts used in discrete geometry. We show that this class is closed under two natural transformations. The first one is called conjugacy and is obtain
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