2 research outputs found
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
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 -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 -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