# Indiscernible extraction and Morley sequences

This is the title of a talk given at the 15th Annual Graduate Student Conference in Logic (University of Wisconsin - Madison) on Apr. 27, 2014. The talk presents the corresponding paper.

## Abstract

It is well known that inside a model of a stable first-order
theory, any long-enough sequence contains indiscernibles. This
fails if the theory is unstable, but Shelah observed that one
can still, in a technical sense, extract indiscernibles on the
side of the original sequence. It turns out such indiscernible
extraction theorems can be used to build Morley sequences, one
of the basic tools of first-order classification theory.

In the unstable case, the lengths involved are quite big: Shelah
asked for the original sequence to have size ב_{γ},
where γ = (2^{|T|})^{+}, and |T| is the number
of formulas in the underlying language. Grossberg, Iovino and
Lessmann later improved this to ב_{δ}, for δ < γ, and asked whether, at least in simple unstable
theories, Morley sequences could be built without using such big
cardinals.

Using a 47 year old idea of Gaifman, I will answer this question by
showing (in simple theories) how to construct a Morley sequence from
any infinite independent sequence. I aim to make the talk reasonably
self-contained and use only minimal background.

## References

- Sebastien Vasey,
*Indiscernible extraction and Morley sequences*, Submitted. Preprint: pdf arXiv.
- Rami Grossberg, José Iovino, Olivier Lessmann.
*A primer of simple theories*, Archive for Mathematical Logic 41 (2002), no. 6, 541-580.
- Hans Adler.
*Thorn-forking as local forking*, Journal of Mathematical Logic 9 (2009), no. 1, 21--38.