Logic Seminar
Scott Mutchnik
UIC
Strict order approximations in hereditary classes
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.
Tuesday September 2, 2025 at 3:00 PM in 636 SEO