69 research outputs found
Tree indiscernibilities, revisited
We give definitions that distinguish between two notions of indiscernibility
for a set \{a_\eta \mid \eta \in \W\} that saw original use in \cite{sh90},
which we name \textit{\s-} and \textit{\n-indiscernibility}. Using these
definitions and detailed proofs, we prove \s- and \n-modeling theorems and
give applications of these theorems. In particular, we verify a step in the
argument that TP is equivalent to TP or TP that has not seen
explication in the literature. In the Appendix, we exposit the proofs of
\citep[{App. 2.6, 2.7}]{sh90}, expanding on the details.Comment: submitte
Generalized Indiscernibles as Model-complete Theories
We give an almost entirely model-theoretic account of both Ramsey classes of
finite structures and of generalized indiscernibles as studied in special cases
in (for example) [7], [9]. We understand "theories of indiscernibles" to be
special kinds of companionable theories of finite structures, and much of the
work in our arguments is carried in the context of the model-companion. Among
other things, this approach allows us to prove that the companion of a theory
of indiscernibles whose "base" consists of the quantifier-free formulas is
necessarily the theory of the Fraisse limit of a Fraisse class of linearly
ordered finite structures (where the linear order will be at least
quantifier-free definable). We also provide streamlined arguments for the
result of [6] identifying extremely amenable groups with the automorphism
groups of limits of Ramsey classes.Comment: 21 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
Practical and Structural Infinitary Expansions
Given a structure 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
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