Indiscernibles in monadically NIP theories

TL;DR

This paper explores the properties of indiscernibles in monadically NIP theories, providing characterizations, studying distality, and examining interpretability of infinite groups, with implications for model theory and combinatorics.

Contribution

It offers new characterizations of monadic NIP via indiscernibles, connects distality with hereditary classes, and shows monadically NIP theories do not interpret infinite groups.

Findings

Characterizations of monadic NIP using indiscernibles

Every planar graph admits a distal expansion

No monadically NIP theory interprets an infinite group

Abstract

We prove various results around indiscernibles in monadically NIP theories. First, we provide several characterizations of monadic NIP in terms of indiscernibles, mirroring previous characterizations in terms of the behavior of finite satisfiability. Second, we study (monadic) distality in hereditary classes and complete theories. Here, via finite combinatorics, we prove a result implying that every planar graph admits a distal expansion. Finally, we prove a result implying that no monadically NIP theory interprets an infinite group, and note an example of a (monadically) stable theory with no distal expansion that does not interpret an infinite group.