## Bootcamp 1 – 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 introduced some errors; email us or comment below and we will happily fix them.

Title: Bootcamp 1 – Informal meeting.

Lecturer: Jaroslav Nešetřil.

Date: September 20, 2016.

Main Topics: Overview over the topics of the DocCourse; classical result in Ramsey theory

Definitions: Arrow notation, Ramsey numbers, arithmetical progression

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

## 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 3Bootcamp 4Bootcamp 5Bootcamp 6Bootcamp 7Bootcamp 8

## 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 1Lecture 2 – Lecture 3

## Introduction to the KPT Correspondence 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: 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

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.

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## 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

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## 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

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## 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,

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## 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

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## 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

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