Practical and Structural Infinitary Expansions

TL;DR

This paper introduces infinitary logic expansions that preserve key properties like the Embedding Ramsey Property and the Modelling Property, leading to stronger results in structural Ramsey theory and ordered structures.

Contribution

It develops a new class of infinitary logic expansions that unify and strengthen existing results in Ramsey theory and model theory, removing technical assumptions.

Findings

Infinitary expansions preserve and reflect ERP and MP.

Every structure with ERP admits a linear order as a union of quantifier-free types.

Strengthened Kechris-Pestov-Todorcevic correspondence for finitary-ERP.

Abstract

Given a structure $M$ we introduce infinitary logic expansions, which generalise the Morleyisation. We show that these expansions are tame, in the sense that they preserve and reflect both the Embedding Ramsey Property (ERP) and the Modelling Property (MP). We then turn our attention to Scow's theorem connecting generalised indiscernibles with Ramsey classes and show that by passing through infinitary logic, one can obtain a stronger result, which does not require any technical assumptions. We also show that every structure with ERP, not necessarily countable, admits a linear order which is a union of quantifier-free types, effectively proving that any Ramsey structure is ``essentially'' ordered. We also introduce a version of ERP for classes of structures which are not necessarily finite (the finitary-ERP) and prove a strengthening of the Kechris-Pestov-Todorcevic correspondence for this notion.