Title: Composing notions of size and their algebraic characterizations
Abstract: One of the earliest results that form part of the Ramsey theory folklore is an observation of Kakeya and Morimoto that subsets of the positive integers with "bounded gaps" (a notion of size) contain arbitrarily long arithmetic progressions. This observation is an instance of what we'll call the Ramsey theory heuristic: underlying many Ramsey theoretic phenomena is (often several) notions of size which contain enough structure to imply an interesting combinatorial pattern. Motivated by this classical result and its modern descendants, we'll state generalizations for sets with bounded gaps and a related dual notion and show how these notions can be "composed"' to produce both new and old notions of size. Composing these notions of size naturally raises a classification problem that roughly asks how the various compositions are related. We'll illustrate these notions via a suggestive visualization and, answering a few instances of the classification problem, sketch the proof of their algebraic characterizations.
(Based on a joint project with John (Cory) Christopherson and Florian Richter.)
