Minicourse on Nonstandard methods in Ramsey theory and combinatorial number theory
Beijing Jiaotong University
Lecturer: Isaac Goldbring
July 15-18, 2024
Abstract:
Nonstandard analysis is the mathematical discipline that provides a rigorous foundation for the use of infinitesimal and infinite numbers, and, more generally, idealized elements of mathematical structures. The subject was initiated by Abraham Robinson, using tools from model theory, although more axiomatic treatments of the subject can be given that require less knowledge of logic. Since its inception, the techniques of nonstandard analysis have been used to provide slicker, conceptually clearer proofs of pre-existing results as well as to prove new results in a wide variety of areas of mathematics, such as functional analysis, probability theory, mathematical physics, and mathematical economics.
In this course, we will focus on applications of nonstandard analysis to Ramsey theory and combinatorial number theory, following closely certain portions of the book Nonstandard methods in Ramsey theory and combinatorial number theory that I wrote with Mauro DiNasso and Martino Lupini. While the book covers a wide range of topics, we will focus on three main themes: extremal/structural graph theory (namely the Triangle removal lemma and the Szemeredi regularity lemma), Jin’s sumset phenomenon (which is arguably one of the earliest applications of nonstandard methods to combinatorial number theory), and the use of iterated nonstandard extensions and idempotent elements (which we use to prove Hindman’s theorem and the Infinite Hales-Jewett theorem).
No previous knowledge of logic, nonstandard analysis, or combinatorics will be assumed. However, prior exposure to measure theory will be very helpful.
Tentative Schedule
- Introduction to nonstandard analysis
- We introduce the basic framework for doing nonstandard analysis and explore the structure of the hyperreal and hypernatural numbers. We introduce the notions of internal and hyperfinite sets and apply them to develop the Loeb measure construction, which will be applied in the two lectures that follow.
- Triangle removal and Szemeredi regularity
- We give nonstandard proofs of two fundamental results from graph theory, the Triangle removal lemma and the Szemeredi regularity lemma. Both results are proven by establishing a hyperfinite version of the result, which is proven using the Loeb measure construction introduced in Lecture 1.
- We prove a theorem of Jin stating that the sum of two positive density subsets of the natural numbers is piecewise syndetic, which is a combinatorial notion of largeness. The theorem is actually a special case of a more general nonstandard result, which also yields a nonstandard proof of Steinhaus’ theorem that the sum of two sets of reals of positive Lebesgue measure contains an interval. Once again, the Loeb measure construction is the key nonstandard tool.
- Idempotents, Hindman’s theorem, and the Hales-Jewett theorem
- We introduce the concept of iterated nonstandard extensions, which allow one to define the notion of an idempotent element of the nonstandard extension of a semigroup. We apply this concept to give simple nonstandard proofs of two gems of Ramsey theory, namely Hindman’s theorem and the Infinite Hales-Jewett theorem.
Other References/Suggested Reading