Graph-like Distributions and Types in Ultrapowers

TL;DR

This paper explores types in ultraproducts with graph-like distributions, providing a set-theoretic characterization and new insights into ultrafilter properties related to graph structures in ultrapowers.

Contribution

It introduces a set-theoretic framework for understanding distributions of types in ultraproducts and offers a novel description of good ultrafilters using graph-theoretic concepts.

Findings

Set-theoretic description of type distributions in ultraproducts.

Characterization of good ultrafilters via graph substructures.

Connection between types, ultrafilters, and graph properties.

Abstract

We study types that appear in ultraproducts that have distributions which can be thought of as a sequence of graphs. The property of having distributions that are captured by graphs is motivated by a commonality of $\mathrm{SOP}_2$-types and types corresponding to pre-cuts in ultrapowers of linear orders. Through this study, we come to a simple set-theoretic description of the distributions for these types and to a new description of good ultrafilters in terms of the difference between "small" external complete subgraphs and internal complete subgraphs in ultraproducts of particular families of finite graphs arising from the comparability relation on the complete binary tree or the intersection of intervals in an infinite linear order.