# On positive local combinatorial dividing-lines in model theory

## TL;DR

This paper introduces positive local combinatorial dividing-lines in model theory, characterizing them via indecomposable Fraisse classes and prime filter classes, and explores their connections with indiscernible collapse and hypergraph classes.

## Contribution

It defines a new class of dividing-lines in model theory and establishes their equivalence with algebraically trivial Fraisse classes and prime filter classes.

## Key findings

- Characterization of positive local combinatorial dividing-lines
- Equivalence with indecomposable algebraically trivial Fraisse classes
- Connections with collapse-of-indiscernibles dividing-lines

## Abstract

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