Quasiminimal Structures and Excellence

This is the title of a three-parts talk I gave at the CMU Model Theory Seminar on Oct. 14, 21, 28, 2013. I would like to thank Rami Grossberg and Alexei Kolesnikov for helpful discussions when preparing this seminar.


A recurring theme in model theory is upward transfer of a property, using only information about countable structures to derive information on all uncountable structures. For example, knowing that an Abstract Elementary Class has no maximal countable model lets us deduce it has an uncountable model. A generalization of this fact was proved by Shelah in a milestone paper, where he showed how to obtain existence of models of an Lω1 sentence at ℵn using a property of systems of countable models indexed by n-dimensional cubes that he called n-goodness. A class where n-goodness holds for all n is called excellent. It turns out that excellence implies existence of arbitrarily large models, and much more.

The concept of excellence got renewed attention, when Boris Zilber used it to study Schanuel's conjecture for the complex numbers. Zilber studied quasiminimal classes that had to satisfy several requirements, including a version of excellence. Until recently, it was unclear whether the excellence requirement was necessary. In this series of talks, I will present a result of Bays, Hart, Hyttinen, Kesälä and Kirby that shows it is not: Excellence holds for free in quasiminimal classes. I aim to make the talks reasonably self-contained and use only minimal backgroud.


Note that the version of Quasiminimal structures and excellence on arXiv is not the latest one. In particular, the proof of Lemma 5.2 in the version linked above is incomplete.