Monadic stability and growth rates of $\omega$-categorical structures

TL;DR

This paper explores the growth rates of omega-categorical and stable structures, revealing a spectrum gap in monadic stability and confirming longstanding conjectures about growth rate classifications.

Contribution

It establishes a connection between monadic stability and a gap in growth rate spectra, providing a nearly complete classification of slower than exponential growth rates.

Findings

Monadic stability corresponds to a gap in growth rate spectrum.

Confirmed longstanding conjectures on growth rate classifications.

Proved the existence of previously unrecognized gaps in growth rates.

Abstract

For $M$ $\omega$-categorical and stable, we investigate the growth rate of $M$, i.e. the number of orbits of $Aut(M)$ on $n$-sets, or equivalently the number of $n$-substructures of $M$ after performing quantifier elimination. We show that monadic stability corresponds to a gap in the spectrum of growth rates, from slower than exponential to faster than exponential. This allows us to give a nearly complete description of the spectrum of slower than exponential growth rates (without the assumption of stability), confirming some longstanding conjectures of Cameron and Macpherson and proving the existence of gaps not previously recognized.