High-Dimensional Ramsey Theory – Course Notes – Spring 2014 – Mike Pawliuk

I am currently taking Stevo Todorcevic’s course (MAT1435HS Topics in Geometric Topology: High-Dimensional Ramsey Theory) at the University of Toronto in Spring 2014. I will be typing up notes and posting them here. Please contact me (by commenting below, or by email) to give me any feedback (typos, questions, clarifications, etc.)

The course is Tuesdays and Thursdays 3:00-4:30 in BA 6180.

Class 1 – Ramsey’s Theorems

Notes for Class 1 [PDF]

Here we look at the classical Ramsey’s Theorems from 1930. We prove the infinite version (of the classical theorem) along the way to proving Ramsey’s theorem that there are “only” 8 binary relations on $\mathbb{N}$ . We then prove the finite version of Ramsey’s Theorem from the infinite version by using an ultraproduct.

Relevant sections from “Introduction to Ramsey Spaces”: 1.1 (Coideals), 1.2 (Dimensions in Ramsey Theory)

Class 2 – Nash-Williams Theorem

Notes for Class 2 [PDF]

We introduce the notion of combinatorial forcing in order to prove that every Nash-Williams family is a Ramsey family. The main idea is to look at combinatorial properties of finite subsets of $\mathbb{N}$ . Furthermore, we look at the notion of blocks and barriers to discuss an analogous theorem of Galvin.

Relevant sections from “Introduction to Ramsey Spaces”: 1.3 (Higher Dimensions in Ramsey Theory)

Class 3 – Partitions of Infinite Dimensional Ramsey Spaces

Notes for Class 3 [PDF]

The main object that we study here is $\mathbb{N}^{[\infty]}$ , the infinite subsets of $\mathbb{N}$ . Given the Ellentuck topology, this is an infinite dimensional Ramsey space. The main result involves investigating (and sometimes equating) the notions of open, meager, Ramsey and nowhere dense in this topology.

Relevant sections from “Introduction to Ramsey Spaces”: 1.4 Ramsey Property and Baire Property.

Class 4 – Left-ideals of Semigroups

Notes for Class 4 [PDF]

We introduce left-ideals of semigroups and point out a handful of straightforward properties of them. This is used to prove the Galvin-Glazer theorem, from which Hindman’s theorem follows immediately.

Relevant sections from “Introduction to Ramsey Spaces”: 2.1 Idempotents in Compacts Semigroups, 2.2. The Galvin Glazer Theorem.

Class 5 – Gowers’ Theorem

Notes for Class 5 [PDF]

This is our first Ramsey space that we examine. It involves looking at block subspaces of FIN $_k$ .

Relevant sections from “Introduction to Ramsey Spaces”: 2.3 Gowers’ Theorem, 2.4 A semigroup of subsymmetric ultrafilters.

Class 6 – The Hales-Jewett Theorem

Notes for Class 6 [PDF]

Here we examine the Hales-Jewett space, which is our second Ramsey space. This one involves looking at finite words (potentially with variable letters).

Relevant sections from “Introduction to Ramsey Spaces”: 2.5 The Hales-Jewett Theorem.

Class 7 – The Halpern-Laüchli Theorem, Part 1

Notes for Class 7 [PDF]

We investigate various different (equivalent) formulations of the Halpern-Laüchli Theorem which is a Ramsey theorem for colourings of finite products of trees.

Relevant sections from “Introduction to Ramsey Spaces”: 3.1 “Versions of the Halpern-Laüchli Theorem.”

Class 8 – The Halpern-Laüchli Theorem, Part 2

Notes for Class 8 [PDF]

We complete the proof of the Halpern-Laüchli Theorem by gluing together some lemmas and various forms of the HL theorem. Some metric versions of HL are discussed.

Relevant sections from “Introduction to Ramsey Spaces”: 3.2 “A Proof of the Halpern-Laüchli Theorem.”

Class 9 – Axioms for an Abstract Ramsey Space

Notes for Class 9 [PDF]

We list the four axioms needed for an Abstract Ramsey Space, motivated by the prototypical Ramsey space: the Hales-Jewett space.

Relevant sections from “Introduction to Ramsey Spaces”: 4.2 The Abstract Ramsey Theorem.

Class 10 – The Abstract Ramsey Theorem, Definitions

Notes for Class 10 [PDF]

We present all of the relevant definitions used in the proof of the ART. Things like $\S$ -Ramsey, $\S$ -Baire and combinatorial forcing are defined.

Relevant sections from “Introduction to Ramsey Spaces”: 4.2 The Abstract Ramsey Theorem.

Class 12 – The Abstract Ramsey Theorem, Proof

Notes for Class 12 [PDF]

This is where most of the work is done for proving the ART. A series of 6 lemmas are proved, mostly by fusion arguments, on the way to proving the big theorem.

Relevant sections from “Introduction to Ramsey Spaces”: 4.3 Combinatorial Forcing.

Class 13 – Title TBA

Notes for Class 13 [PDF] – SOON

Summary

Relevant sections from “Introduction to Ramsey Spaces”:

Class 14 – Title TBA

Notes for Class 14 [PDF] – SOON