  We study a conjecture linking ultimate periodicity of infnite words to the
existence of colorings on finite words avoiding monochromatic factorisation of
suffixes, with the extra condition that the ordered concatenation of elements
of this factorisation remains monochromatic. This type of results shows the
limits of Ramsey theory in the context of combinatorics on words. We show some
reductions of the problem and the example of the Zimin word. Using the new
notion of consecutive length, we show that words avoiding large squares fulfill
the conjecture.
