# The number of models of a fixed Scott rank, for a counterexample to the analytic Vaught conjecture

## TL;DR

The paper investigates the number of models with fixed Scott rank in counterexamples to the analytic Vaught conjecture, showing a club set where the number of models remains constant at each rank.

## Contribution

It establishes a new structural property of counterexamples to the analytic Vaught conjecture regarding the distribution of models across Scott ranks.

## Key findings

- Existence of a club set where the number of models of each Scott rank is constant.
- Counterexamples with a fixed number of models at Scott rank ω₁ have a predictable distribution across a club set.
- Provides insight into the structure of models in the context of the analytic Vaught conjecture.

## Abstract

We show that if $\gamma \in \omega \cup \{\aleph_{0}\}$ and $\mathcal{A}$ is a counterexample to the analytic Vaught conjecture having exactly $\gamma$ many models of Scott rank $\omega_{1}$, then there exists a club $C \subseteq \omega_{1}$ such that $\mathcal{A}$ has exactly $\gamma$ many models of Scott rank $\alpha$, for each $\alpha \in C$.