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.


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 ב‎γ, where γ = (2|T|)+, and |T| is the number of formulas in the underlying language. Grossberg, Iovino and Lessmann later improved this to ב‎δ, 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.