A factorisation x=u1u2⋯ 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 uk1uk2⋯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 exist 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
Seminar series
Date
Tue, 13 Jun 2023
Time
14:00 -
15:00
Location
L5
Speaker
Maria Ivan
Organisation
University of Cambridge