Topological dynamics and Ramsey classes – Ramsey DocCourse Prague 2016

The following notes are from the Ramsey DocCourse in Prague 2016. The notes are taken by me and I have edited them. In the process I may have introduced some errors; email me or comment below and I will happily fix them.

Title: Topological dynamics and Ramsey classes.

Lecturer: Lionel Ngyuen Van Thé.

Date: November 14, 2016.

Main Topics: Proof of KPT correspondence between extreme amenability and ramsey class.

Definitions: Topological group, $S_\infty$, $d_R, d_L$, Polish group, ultrametric, $G$-flow, extreme amenability.

Fractional Hedetniemi’s conjecture and Chromatic Ramsey number – Ramsey DocCourse Prague 2016

The following notes are from the Ramsey DocCourse in Prague 2016. The notes are taken by me and I have edited them. In the process I may have introduced some errors; email me or comment below and I will happily fix them.

Title: Fractional Hedetniemi’s conjecture and Chromatic Ramsey number

Lecturer: Xuding Zhu

Date: November 9, 2016

Main Topics: Chromatic Ramsey numbers, lower bound for them, Hedetniemi’s conjecture, fractional Hedetniemi’s conjecture.

Definitions: $\rho$-Ramsey number, $\chi$-Ramsey number, wreath product, product graph, graph homomorphism, fractional chromatic number

The first dynamical system; Random Number Theory – Ramsey Doccourse Prague 2016

The following notes are from the Ramsey DocCourse in Prague 2016. The notes are taken by me and I have edited them. In the process I may have introduced some errors; email me or comment below and I will happily fix them.

Title: The first dynamical system; and Random Number Theory

Lecturer: Carl Pomerance

Date: November 8, 2016

Main Topics: Chains with $\sigma$, distribution of primes, randomness in math

Definitions: Amicable, Perfect, Abundant, Deficient

Hrushovski constructions – Ramsey DocCourse Prague 2016

The following notes are from the Ramsey DocCourse in Prague 2016. The notes are taken by me and I have edited them. In the process I may have introduced some errors; email me or comment below and I will happily fix them.

Title: Hrushovski constructions 1 (of 3)

Lecturer: David Evans

Date: November 7, 2016

Main Topics: Definition Review, $k$-very-sparse iff $k$-orientable, Existence of graph without Ramsey expansion.

Definitions: $k$-very-sparse, $k$-orientable,

Dynamical systems and Ramsey theory – Ramsey DocCourse Prague 2016

The following notes are from the Ramsey DocCourse in Prague 2016. The notes are taken by me and I have edited them. In the process I may have introduced some errors; email me or comment below and I will happily fix them.

Title: Ergodic theory and amenable groups

Lecturer: Benjamin Weiss

Date: October 31, 2016.

Main Topics: Three equivalent notions of amenability, Basic concepts in ergodic actions, Furstenberg’s Ergodic proof of Szemerédi’s theorem

Definitions: Ergodic action, weak mixing, mixing, Banach limit, amenable group, left invariant mean, paradoxical decomposition, Følner sequence

Partite constructions 2 – Ramsey DocCourse Prague 2016

The following notes are from the Ramsey DocCourse in Prague 2016. The notes are taken by me and I have edited them. In the process I may have introduced some errors; email me or comment below and I will happily fix them.

Title: Partite consturctions 2 (of 3)

Lecturer: Jaroslav Nešetril

Date: Tuesday October 25, 2016.

Main Topics: There are graphs with large chromatic number but no small cycles, Tutte’s construction, Edge-Ramsey for graphs (using partite construction)

Definitions: No New definitions.

Part 1 – Part 2 – Part 3

Ramsey and Ultrafilters 2 – Ramsey DocCourse Prague 2016

The following notes are from the Ramsey DocCourse in Prague 2016. The notes are taken by me and I have edited them. In the process I may have introduced some errors; email me or comment below and I will happily fix them.

Special thanks to Ivan Khatchatourian for some elaborations.

Title: Ramsey and Ultrafilters 2 (of 2)

Lecturer: Slawomir Solecki

Date: Friday October 21, 2016.

Main Topics: Abstraction of Gowers/Lupini/Furstenberg-Katznelson, worked example with Furstenberg-Katznelson, slides about tensors

Definitions: Forestation of a Poset, Semigroup generated by a poset, [Specific to these constructions], $\hat{Y^{\prime\prime}(M)}$

Lecture 1 – Lecture 2

These two lectures are based on the following 2016 paper of Solecki: “Monoid actions and ultrafilter methods in Ramsey theory”.
Continue reading Ramsey and Ultrafilters 2 – Ramsey DocCourse Prague 2016