Indiscernibles in monadically NIP theories
Samuel Braunfeld, Michael C. Laskowski
TL;DR
This work develops a unified understanding of indiscernibles in monadically NIP theories, providing equivalent characterizations via indiscernible widening and dp^+-minimality, and clarifying how a single singleton can interact with an indiscernible sequence. It then analyzes distality in the monadic setting, showing a precise collapse: monadic distality is equivalent to distality together with monadic NIP, and establishing distal expansions for hereditary classes with bounded twin-width, including planar graphs. A key algebraic consequence is that monadically NIP theories cannot trace-define an infinite group, with broader implications for the presence of algebraic structure in such theories. The paper also presents a stable-theory example without distal expansion that avoids interpreting an infinite group, illustrating delicate interactions between stability, distality, and trace definability in the monadic realm.
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.