A Ramsey characterisation of eventually periodic words

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