Failure on n-uniqueness: a family of examples

TL;DR

This paper explores the limitations of n-uniqueness in stable theories by constructing examples that demonstrate failure at certain levels of n-amalgamation, linking model theory with permutation groups.

Contribution

It introduces a family of stable theories that exhibit specific failures of (n+1)-existence and (n+1)-uniqueness, generalizing previous examples for n=2.

Findings

Existence of stable theories with (k+1)-existence and k-uniqueness for all k<n+1

Failure of (n+2)-existence and (n+1)-uniqueness in these theories

Generalization of Hrushovski's example for n=2

Abstract

In this paper, the connections between model theory and the theory of infinite permutation groups are used to study the n-existence and the n-uniqueness for n-amalgamation problems of stable theories. We show that, for any n>1, there exists a stable theory having (k+1)-existence and k-uniqueness, for every k<n+1, but that does not have neither (n+2)-existence nor (n+1)-uniqueness. In particular, this generalizes the example, for n=2, due to E.Hrushovski given in [3].