Shearing in some simple rank one theories

TL;DR

This paper introduces and characterizes shearing, a new notion related to inconsistency along generalized indiscernible sequences, to analyze simple theories and distinguish between different classes of unstable rank one theories.

Contribution

It defines shearing, characterizes it for the random graph, and uses it to differentiate between the random graph and higher-order triangle-free analogues, showing shearing's distinctness from dividing.

Findings

Shearing characterizes the random graph.

Shearing distinguishes between the random graph and $T_{n,k}$ theories.

Shearing is shown to be different from dividing in simple unstable theories.

Abstract

Dividing asks about inconsistency along indiscernible sequences. In order to study the finer structure of simple theories without much dividing, the authors recently introduced shearing, which essentially asks about inconsistency along generalized indiscernible sequences. Here we characterize the shearing of the random graph. We then use shearing to distinguish between the random graph and the theories $T_{n,k}$, the higher-order analogues of the triangle-free random graph. It follows that shearing is distinct from dividing in simple unstable theories, and distinguishes meaningfully between classes of simple unstable rank one theories. The paper begins with an overview of shearing, and includes open questions.