This course delves into the Ramsey theory of homogeneous structures, exploring the challenges and breakthroughs in extending Ramsey theory from finite to infinite structures. The course covers topics such as Ramsey's Theorems, the Ramsey theory of the rationals and the Rado graph, and methodologies for discovering Ramsey properties of homogeneous structures. The teaching method involves discussing structural properties, ongoing research, and the set-theoretic method of forcing. The course is designed for individuals interested in advanced topics in mathematics, particularly in Ramsey theory and infinite-dimensional structures.
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
