4,370 research outputs found
Ramsey precompact expansions of homogeneous directed graphs
In 2005, Kechris, Pestov and Todorcevic provided a powerful tool to compute
an invariant of topological groups known as the universal minimal flow,
immediately leading to an explicit representation of this invariant in many
concrete cases. More recently, the framework was generalized allowing for
further applications, and the purpose of this paper is to apply these new
methods in the context of homogeneous directed graphs.
In this paper, we show that the age of any homogeneous directed graph allows
a Ramsey precompact expansion. Moreover, we verify the relative expansion
properties and consequently describe the respective universal minimal flows
On Ramsey properties of classes with forbidden trees
Let F be a set of relational trees and let Forbh(F) be the class of all
structures that admit no homomorphism from any tree in F; all this happens over
a fixed finite relational signature . There is a natural way to expand
Forbh(F) by unary relations to an amalgamation class. This expanded class,
enhanced with a linear ordering, has the Ramsey property.Comment: Keywords: forbidden substructure; amalgamation; Ramsey class; partite
method v2: changed definition of expanded class; v3: final versio
The strength of Ramsey Theorem for coloring relatively large sets
We characterize the computational content and the proof-theoretic strength of
a Ramsey-type theorem for bi-colorings of so-called {\em exactly large} sets.
An {\it exactly large} set is a set X\subset\Nat such that
\card(X)=\min(X)+1. The theorem we analyze is as follows. For every infinite
subset of \Nat, for every coloring of the exactly large subsets of
in two colors, there exists and infinite subset of such that is
constant on all exactly large subsets of . This theorem is essentially due
to Pudl\`ak and R\"odl and independently to Farmaki. We prove that --- over
Computable Mathematics --- this theorem is equivalent to closure under the
Turing jump (i.e., under arithmetical truth). Natural combinatorial
theorems at this level of complexity are rare. Our results give a complete
characterization of the theorem from the point of view of Computable
Mathematics and of the Proof Theory of Arithmetic. This nicely extends the
current knowledge about the strength of Ramsey Theorem. We also show that
analogous results hold for a related principle based on the Regressive Ramsey
Theorem. In addition we give a further characterization in terms of truth
predicates over Peano Arithmetic. We conjecture that analogous results hold for
larger ordinals
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