Talk given at IST Austria on January 20, 2020. Here are the slides.
Ramsey's theorem says that for each natural number n, there exists a natural number N so that each graph with N vertices contains either a clique or an independent set of size n. A theorem of Erdős and Rado generalizes it to infinite cardinals. Ramsey himself showed that one can take n = N if n is the first infinite cardinal but in most other uncountable cases N must be much bigger than n. Stability theory is a branch of model theory studying certain definability conditions allowing us to take n = N for a large number of infinite cardinals. Historically, stability theory was first developed by Shelah for classes axiomatized by first-order formulas. In this talk, I will describe a generalization to a large class of concrete categories: abstract elementary classes. I will also talk about recent progresses on the field's main test question, the eventual categoricity conjecture, resolved by Morley and Shelah for first-order but still open for abstract elementary classes.
Sebastien Vasey, Accessible categories, set theory, and model theory: an invitation, Preprint: pdf arXiv, 68 pages.
Michael Lieberman, Jiří Rosický, and Sebastien Vasey, Forking independence from the categorical point of view, Advances in Mathematics 346 (2019), 719–772. Publisher version pdf arXiv.
Sebastien Vasey, The categoricity spectrum of large abstract elementary classes, Selecta Mathematica 23 (2019), no. 5, 65 (51 pages). Publisher version pdf arXiv.
Saharon Shelah and Sebastien Vasey, Categoricity and multidimensional diagrams, preprint: pdf arXiv, 63 pages.