1,118 research outputs found
The externally definable Ramsey property and fixed points on type spaces
We discuss the externally definable Ramsey property, a weakening of the
Ramsey property for ultrahomogeneous structures, where the only colourings
considered are those that are externally definable: that is, definable with
parameters in an elementary extension. We show a number of basic results
analogous to the classical Ramsey theory, and show that, for an
ultrahomogeneous structure M, the externally definable Ramsey property is
equivalent to the dynamical statement that, for each natural number n, every
subflow of the space of n-types with parameters in M has a fixed point. We
discuss a range of examples, including results regarding the lexicographic
product of structures.Comment: 42 pages, 1 figur
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
Ramsey Properties of Permutations
The age of each countable homogeneous permutation forms a Ramsey class. Thus,
there are five countably infinite Ramsey classes of permutations.Comment: 10 pages, 3 figures; v2: updated info on related work + some other
minor enhancements (Dec 21, 2012
The Ramsey Theory of Henson graphs
Analogues of Ramsey's Theorem for infinite structures such as the rationals
or the Rado graph have been known for some time. In this context, one looks for
optimal bounds, called degrees, for the number of colors in an isomorphic
substructure rather than one color, as that is often impossible. Such theorems
for Henson graphs however remained elusive, due to lack of techniques for
handling forbidden cliques. Building on the author's recent result for the
triangle-free Henson graph, we prove that for each , the
-clique-free Henson graph has finite big Ramsey degrees, the appropriate
analogue of Ramsey's Theorem.
We develop a method for coding copies of Henson graphs into a new class of
trees, called strong coding trees, and prove Ramsey theorems for these trees
which are applied to deduce finite big Ramsey degrees. The approach here
provides a general methodology opening further study of big Ramsey degrees for
ultrahomogeneous structures. The results have bearing on topological dynamics
via work of Kechris, Pestov, and Todorcevic and of Zucker.Comment: 75 pages. Substantial revisions in the presentation. Submitte
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