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Bootcamp 2 – Ramsey DocCourse Prague 2016

The following notes are from the Ramsey DocCourse in Prague 2016. The notes were taken and edited by myself and Michael Kompatscher. In the process we may have included some errors; email us or comment below and we will happily fix them.

Title: Bootcamp 2 (of 8)

Lecturer: Jaroslav Nešetřil.

Date: September 21, 2016.

Main Topics: The Rado graph, homogeneous structures, universal graphs

Definitions: Language, structures, homomorphisms, embeddings, homogeneity, universality, Rado graph (Random graph),…

Bootcamp 1 – Bootcamp 2 – Bootcamp 3 – Bootcamp 4 – Bootcamp 5 – Bootcamp 6 – Bootcamp 7 – Bootcamp 8

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Introduction to the KPT correspondence 3 – 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: Introduction to the KPT correspondence 3 (of 3).

Lecturer: Lionel Ngyuen Van Thé.

Date: November 18, 2016.

Main Topics:

Definitions: Expansion property,

Lecture 1 – Lecture 2 – Lecture 3

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Introduction to the KPT Correspondence 2 – Ramsey DocCourse Prague 2016

Title: Introduction to the KPT correspondence 2 (of 3).

Lecturer: Lionel Ngyuen Van Thé.

Date: November 16, 2016.

Main Topics: Computing universal minimal flows, $M(S_\infty)$ , why precompactness is important.

Definitions: Minimal flow, universal flow, Logic action, $G$ -equivariant.

Lecture 1 – Lecture 2 – Lecture 3

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Topological dynamics and Ramsey classes – Ramsey DocCourse Prague 2016

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.

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

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

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

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

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Hrushovski constructions – Ramsey DocCourse Prague 2016

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,

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Dynamical systems and Ramsey theory – Ramsey DocCourse Prague 2016

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

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Partite constructions 2 – Ramsey DocCourse Prague 2016

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

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Ramsey and Ultrafilters 2 – Ramsey DocCourse Prague 2016

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”.
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