Higher amalgamation properties in measured structures

TL;DR

This paper proves a higher amalgamation property in certain measurable structures using an infinitary Hypergraph Removal Lemma, distinguishing them from supersimple structures with finite SU-rank, and applies this to analyze Hrushovski's structures.

Contribution

It introduces a new higher amalgamation result for measurable structures using an infinitary combinatorial approach, expanding understanding of model-theoretic properties.

Findings

Independent amalgamation property holds in measurable structures.

Some Hrushovski structures are shown not to be MS-measurable.

The result differentiates measurable structures from supersimple finite SU-rank structures.

Abstract

Using an infinitary version of the Hypergraph Removal Lemma due to Towsner, we prove a model-theoretic higher amalgamation result. In particular, we obtain an independent amalgamation property which holds in structures which are measurable in the sense of Macpherson and Steinhorn, but which is not generally true in structures which are supersimple of finite SU-rank. We use this to show that some of Hrushovski's non-locally-modular, supersimple $\omega$-categorical structures are not MS-measurable.