Scott Mutchnik : Strict order approximations in hereditary classes

Posted by Scott Mutchnik , part of the Logic Seminar.

At

Sept. 2, 2025, 3 p.m.

In

636 SEO

Abstract

It was observed very recently that the classical SOP_n hierarchy, a family of approximations of the strict order property which Shelah introduced with the goal of classifying non-simple first-order theories, extends to a hierarchy of properties SOP_r for r any real number at least 3. However, it remains open whether the real-valued NSOP_r hierarchy is distinct from the original integer-valued NSOP_n hierarchy. To make this question more tractable, we can ask it at the quantifier-free level, obtaining a real-valued quantity, of independent combinatorial interest, associated with any hereditary class of finite structures. While it is also open whether this quantity can have non-integer values, we can show that, in the case of a hereditary class defined by finitely many omitted weak substructures, it is an integer. We will discuss, and aim to prove, this result.