Tag: Ramsey Theory

Euclidean Ramsey Theory 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: Euclidean Ramsey Theory 2 (of 3).

Lecturer: David Conlon.

Date: November 25, 2016.

Main Topics: Ramsey implies spherical, an algebraic condition for spherical, partition regular equations, an analogous result for edge Ramsey.

Definitions: Spherical, partition regular.

Lecture 1 – Lecture 2 – Lecture 3

Ramsey DocCourse Prague 2016 Index of lectures.

Continue reading Euclidean Ramsey Theory 2 – Ramsey DocCourse Prague 2016

Dual Ramsey, an introduction – 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: Dual Ramsey, the Gurarij space and the Poulsen simplex 1 (of 3).

Lecturer: Dana Bartošová.

Date: December 12, 2016.

Main Topics: Comparison of various Fraïssé settings, metric Fraïssé definitions and properties, KPT of metric structures, Thick sets

Definitions: continuous logic, metric Fraïssé properties, NAP (near amalgamation property), PP (Polish Property), ARP (Approximate Ramsey Property), Thick, Thick partition regular.

Lecture 1 – Lecture 2 – Lecture 3

Ramsey DocCourse Prague 2016 Index of lectures.

Continue reading Dual Ramsey, an introduction – Ramsey DocCourse Prague 2016

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

Continue reading Bootcamp 1 – Ramsey DocCourse Prague 2016

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

Continue reading Bootcamp 2 – Ramsey DocCourse Prague 2016

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

Continue reading Introduction to the KPT correspondence 3 – Ramsey DocCourse Prague 2016

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.

Continue reading Topological dynamics and Ramsey classes – Ramsey DocCourse Prague 2016

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

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

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,

Continue reading Hrushovski constructions – Ramsey DocCourse Prague 2016

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

Continue reading Dynamical systems and Ramsey theory – Ramsey DocCourse Prague 2016

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