# Ramsey characterisation of eventually periodic words

# Ramsey characterisation of eventually periodic words

A factorisation x = u_1 u_2 ... of an infinite word x on alphabet X is called `super-monochromatic', for a given colouring of the finite words X* on alphabet X, if each word u_{k_1} u_{k_2} ... u_{k_n}, where k_1 < ... < k_n, 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 exists 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