Generalised Indiscernibles, Dividing Lines, and Products of Structures

Generalised indiscernibles highlight a strong link between model theory and structural Ramsey theory. In this paper, we use generalised indiscernibles as tools to prove results in both these areas. More precisely, we first show that a reduct of an ultrahomogenous $\aleph_0$-categorical structure which has higher arity than the original structure cannot be Ramsey. In particular, the only nontrivial Ramsey reduct of the generically ordered random $k$-hypergraph is the linear order. We then turn our attention to model-theoretic dividing lines that are characterised by collapsing generalised indiscernibles, and prove, for these dividing lines, several transfer principles in (full and lexicographic) products of structures. As an application, we construct new algorithmically tame classes of graphs

Practical and Structural Infinitary Expansions

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.Comment: 22 pages. Minor corrections and rearrangement of sections. Section 8 of the previous version will appear in a separate pape

Products of Classes of Finite Structures

We study the preservation of certain properties under products of classes of finite structures. In particular, we examine age indivisibility, indivisibility, definable self-similarity, the amalgamation property, and the disjoint n-amalgamation property. We explore how each of these properties interact with the wreath product, direct product, and free superposition of classes of structures. Additionally, we consider the classes of theories which admit configurations indexed by these products.Comment: 33 page

Ranks Based on Algebraically Trivial Fraisse Classes

In this paper, we introduce the notion of K-rank, where K is an algebraically trivial Fraisse class. Roughly speaking, the K-rank of a partial type is the number of "copies" of K that can be "independently coded" inside of the type. We study K-rank for specific examples of K, including linear orders, equivalence relations, and graphs. We discuss the relationship of K-rank to other ranks in model theory, including dp-rank and op-dimension (a notion coined by the first author and C. D. Hill in previous work).Comment: 42 page