Dynamical Systems Seminar: From Ramsey Theory to Combinatorics on Words: How a forgotten proof helped find long arithmetic progressions

Speaker: Petra Staynova, University of Derby

Title: From Ramsey Theory to Combinatorics on Words: How a forgotten proof helped find long arithmetic progressions

Abstract:

The original proofs of classic Ramsey-theoretic results, such as van der Waerden's theorem, have been forgotten and replaced by more elegant and mysterious proofs via ultrafilters and the Stone-Czech compactification of the integers. However, there is still good value to be found in the original proofs. In particular, van der Waerden's proof of his theorem about the existence of arbitrarily long monochromatic arithmetic progressions is both beautifully combinatorial and intuitive. It gives a concrete construction of the arithmetic progressions, and an upper bound on 'how far you have to look before you find a monochromatic arithmetic progression'. We can use van der Waerden's ideas to give an elegant visual proof of a combinatorially difficult result about arithmetic progressions within the Thue-Morse word. Furthermore, we can show that the progression we find via this method is, in some sense, the longest possible. This is joint work with Ibai Aedo, Uwe Grimm, and Yasushi Nagai.