On positive local combinatorial dividing-lines in model theory

Abstract

We introduce the notion of positive local combinatorial dividing-lines in model theory. We show these are equivalently characterized by indecomposable algebraically trivial Fraïssé classes and by complete prime filter classes. We exhibit the relationship between this and collapse-ofindiscernibles dividing-lines. We examine several test cases, including those arising from various classes of hypergraphs.

Products of Classes of Finite Structures

“…There has been significant interest in model theoretic dividing lines that are given by the presence or absence of a certain definable combinatorial objects. In [8] and [10], the authors investigate a general framework for studying these dividing lines called configurations. This led to a natural partial ordering on dividing lines; see Example 5.5.…”

“…The class C K coincides with the class of theories that admit a non-collapsing K-generalized indiscernible, assuming that K is a Fraïssé class with the Ramsey property. For more information on this, see [8] and [9].…”

“…In [8] and [10], the authors study a related approach to examining model theoretic dividing lines using the notion of configurations, which "code" classes of structures into a theory. All of the current dividing lines obtained using configurations coincide with known dividing lines (e.g., stability, NIP, etc.).…”

“…All of the current dividing lines obtained using configurations coincide with known dividing lines (e.g., stability, NIP, etc.). An interesting open question from [8] and [10], restated as Question 5.6 below, is whether there are new dividing lines which can be obtained this way. We study the preservation of configurations under products in an attempt to answer this question.…”