A dependent theory with few indiscernibles

TL;DR

This paper constructs a dependent theory demonstrating that large sets can lack indiscernible sequences unless set-theoretic conditions are met, addressing a fundamental question in model theory.

Contribution

It provides a complete solution to the existence of indiscernibles in dependent theories by constructing a specific theory with precise combinatorial properties.

Findings

Existence of dependent theories with large sets lacking indiscernibles

Characterization of indiscernibility conditions in dependent theories

Connection between set theory and indiscernible sequences

Abstract

We give a full solution to the question of existence of indiscernibles in dependent theories by proving the following theorem: for every $\theta$ there is a dependent theory $T$ of size $\theta$ such that for all $\kappa$ and $\delta$, $\kappa\to\left(\delta\right)_{T,1}$ iff $\kappa\to\left(\delta\right)_{\theta}^{<\omega}$. This means that unless there are good set theoretical reasons, there are large sets with no indiscernible sequences.