Explore Ramsey theory of homogeneous structures in this comprehensive lecture. Delve into the historical development of Ramsey theory on relational structures, from Ramsey's seminal theorem to recent breakthroughs. Understand the challenges in applying finite structure methods to infinite homogeneous counterparts, and discover how set-theoretic forcing opened new pathways in this field. Examine the structural properties involved in the Ramsey theory of various homogeneous structures, including the rationals, Rado graph, and homogeneous triangle-free graph. Learn about ongoing work and related areas such as infinite-dimensional Ramsey theory and partition theory of uncountable structures. Gain insights from the works of numerous mathematicians in this field, and explore the syllabus covering topics from the Pigeonhole Principle to ongoing investigations in binary free amalgamation classes.
Overview
Syllabus
Intro
Pigeonhole Principle
Ramsey's Theorems, 1930
The Rationals as a Dense Linear Order
Coloring Finite Sets of Rationals
Example: Colorings of copies of a finite graph
Homogeneous and Universal Structures
Infinite Structural Ramsey Theory
Big Ramsey Degree results up to 2010
Methodology for red highlighted results
Milliken's Theorem (special case)
Diagonal Antichains
Exact Big Ramsey Degrees of the Rado graph
Prior Methods Insufficient for Triangle-Free Graphs
New developments: how forcing opened new paths
Diagonal Coding Tree T within Coding Tree of 1-types
Forcing a Level Set Pigeonhole
Exact Big Ramsey Degrees for H3
Precise characterization of big Ramsey degrees for Hz
T(Edge, H3) = 2
T(Non-Edge, H3) = 5
Developments via coding trees and forcing
Developments not using forcing
Upper Bounds for Binary Free Amalgamation Classes
Exact Big Ramsey Degrees for binary free amalgamation
Big Question
Ongoing Investigations
Taught by
International Mathematical Union
