Abstract

A factorisation x = u1 u2 · · · of an infinite word x on alpha- bet X is called ‘super-monochromatic’, for a given colouring of the finite words X∗ on alphabet X, if each word uk1 uk2 · · · ukn , where k1 < · · · < kn, is the same colour. A direct application of Hindman’s theorem shows that if x is eventually periodic, then for every finite colouring of X∗, there ex- ist a suffix of x that admits a super-monochromatic factorisation. What about the converse? In this talk we show that the converse does indeed hold: thus 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. This has been a conjecture in the community for some time. Our main tool is a Ramsey result about alternating sums. This provides a strong link between Ramsey theory and the combinatorics of infinite words. Joint work with Imre Leader and Luca Q. Zamboni