Tag: Ramsey Theory

Ramsey and Ultrafilters 1 – 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 clarifying some typos and adding some elaborations.

Title: Ramsey and Ultrafilter 1 (of 2)

Lecturer: Slawomir Solecki

Date: Wednesday October 19, 2016.

Main Topics: Gowers’ Theorem, Lupini’s Theorem, Furstenberg-Katznelson Theorem, Monoids, Semigroups.

Definitions: [many]

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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Bootcamp 8 – 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: Bootcamp 8 (of 8)

Lecturer: Jan Hubička

Date: Wednesday October 12, 2016.

Main Topics: Ramsey lifts, Classification results, Tournaments, Digraphs, Permutations, unary functions, Steiner Systems, Dual Ramsey, Graham-Rothschild.

Definitions: Digraph, Tournament, Interposition, \text{CSP}(G) , Strong substructure, unary function, Steiner System

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

Note: The material here is meant as an overview, so many details are missing.
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Functions on Homogeneous Ramsey Structures 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: Functions on Homogeneous Ramsey Structures 3 (of 3).

Lecturer: Michael Pinsker.

Date: Friday October 14, 2016.

Main Topics: Proof of Ramsey claim for canonical functions, Thomas’ Theorem for the Rado Graph, Open questions

Definitions: No new definitions

Lecture 1 – Lecture 2 – Lecture 3

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Bootcamp 7 – 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: Bootcamp 7 (of 8)

Lecturer: Jan Hubička

Date: Monday October 10, 2016.

Main Topics: “Correct” definition of Ramsey expansion, Ramsey lifts/expansions of graphs, Ramsey lifts/expansions of posets.

Definitions: Precompact expansion, Expansion property, Free Amalgamation, strong type, equivalence formula, equivalence closure, interpretation in a model.

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

INCOMPLETE: One dimensional proof of t copies of K_\omega , \omega copies of K_\omega .

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Functions on Homogeneous Ramsey Structures 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. Thank you to Michael Kompatscher for all his feedback and helpful comments.

Title: Functions on Homogeneous Ramsey Structures 2 (of 3).

Lecturer: Michael Pinsker.

Date: Friday October 7, 2016.

Main Topics: Equivalent notions of \omega -categorical, Proof of Cameron’s Theorem, Proof of canonical-Ramsey.

Definitions:No new ones.

Lecture 1 – Lecture 2 – Lecture 3

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Bootcamp 6 – 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: Bootcamp (6 of 6)

Lecturer: Jaroslav Nesetril

Date: Monday October 3, 2016.

Main Topics: Other applications of the “product argument”, Chain-Ramsey for Posets, Proof of edge-Ramsey for Graphs, Proof of Hales-Jewett

Definitions: Structural pigeonhole principle, Poset, Graph product, Combinatorial line

Bootcamp 1 – Bootcamp 2 – Bootcamp 3 – Bootcamp 4 – Bootcamp 5 – Bootcamp 6
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Bootcamp 5 – 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: Bootcamp 5 (of 8)

Lecturer: Jan Hubička

Date: Friday September 30, 2016.

Main Topics: Rado Graph, Fraïssé’s Theorem, Examples of Fraïssé classes, Ramsey implies Amalgamation, Lifts and Reducts, Ramsey classes have linear orders

Definitions: Extension Property, Ultrahomogeneous, Universal, \text{Age}(A) , Fraïssé class, irreducible structure, Lifts/Expansions and Shadows/reducts.

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

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Bootcamp 4 – 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 Matěj Konečný for helpful feedback.

Title: Bootcamp 4 (of 8)

Lecturer: Jaroslav Nešetřil

Date: Monday September 26, 2016.

Main Topics: Graphs have the ordering property, Ordered Edge-Ramsey implies Ordering property, Graphs are not Ramsey, Amalgamation for Ramsey classes, the product argument, Hilbert’s Theorem.

Definitions: Joint Embedding Property, Amalgamation Property, statement of Hilbert’s Theorem.

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

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Bootcamp 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: Bootcamp (3 of 6)

Lecturer: Jaroslav Nešetřil

Date: Friday September 23, 2016.

Main Topics: Point Ramsey for graphs, \langle A,B,C \rangle hypergraphs.

Definitions: Ramsey property for finite structures, Ramsey Class, point-Ramsey, edge-Ramsey, \langle A,B,C \rangle hypergraphs, Chromatic number, Ordering Property.

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

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High-Dimensional Ramsey Theory – Course Notes – Spring 2014

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.)

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