Date
Tue, 13 Jun 2023
Time
14:00 - 15:00
Location
L5
Speaker
Maria Ivan
Organisation
University of Cambridge

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 uk1uk2ukn, 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

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