Infinite Ramsey theory

responsible:  Aleksandra Kwiatkowska

time and place: Mondays at 10:15, place SR1d

The first seminar will be on 8.04.

Announcement

Hindman’s theorem states that for any colouring of natural numbers into finitely many colours there is an infinite set whose all finite sums are in a single colour. This can be shown using ultrafilters and the Ellis’ theorem on the existence of idempotents in compact semi- groups. In the seminar we will prove this and many other infinite Ramsey theorems, such as Gowers’s theorem, which gave a positive solution to the distortion problem for the Banach space c_0, as well as infinite versions of the Hales-Jewett theorem. We will prove Ellentuck’s theorem on completely Ramsey sets, which was used by Rosenthal to characterize Banach spaces in which the space \ell_1 embeds.

Prerequisites: basic knowledge of logic, general topology, and analysis.

Topics to cover

1) Introduction. A couple of proofs of van der Waerden theorem (speaker: A. Kwiatkowska). notes

2) Polynomial van der Waerden theorem (speaker: R. Sullivan). notes

3) Compact topological semigroups, idempotents, Hindman's theorem (speaker: S. Kawamoto). notes

4) Gower's theorem for FIN_k (speaker: J. Pietsch). notes

5) Infinite Hales-Jewett

6-8+) Sections 1-4 of paper 2)

10) Galvin-Prikry and Ellentuck topology

References

1) A. Kechris, Classical Descriptive Set Theory, Springer; 1995th edition.
2) M. Lupini, Actions on semigroups and an infinitary Gowers-Hales-Jewett Ramsey theorem, Sections 1-4, Trans. Amer. Math. Soc.371(2019), no.5, 3083–3116.

3) T. Tao blog post https://terrytao.wordpress.com/tag/hindmans-theorem/

4) S. Todorcevic, Introduction to Ramsey spaces, Chapter 2, Ann. of Math. Stud., 174 Princeton University Press, Princeton, NJ, 2010, viii+287 pp.