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1. CMB 2012 (vol 56 pp. 709)
Universal Minimal Flows of Groups of Automorphisms of Uncountable Structures It is a well-known fact, that the greatest ambit for
a topological group $G$ is the Samuel compactification of $G$ with
respect to the right uniformity on $G.$ We apply the original
description by Samuel from 1948 to give a simple computation of the
universal minimal flow for groups of automorphisms of uncountable
structures using FraÃ¯ssÃ© theory and Ramsey theory. This work
generalizes some of the known results about countable structures.
Keywords:universal minimal flows, ultrafilter flows, Ramsey theory Categories:37B05, 03E02, 05D10, 22F50, 54H20 |
2. CMB 2011 (vol 55 pp. 410)
A Ramsey Theorem with an Application to Sequences in Banach Spaces The notion of a maximally conditional sequence is introduced for sequences in a Banach space. It is then proved using
Ramsey theory that every basic sequence in a Banach space has a subsequence which is either an unconditional
basic sequence or a maximally conditional sequence. An apparently novel, purely combinatorial lemma in the spirit of
Galvin's theorem is used in the proof. An alternative proof
of the dichotomy result for sequences in Banach spaces is
also sketched,
using the Galvin-Prikry theorem.
Keywords:Banach spaces, Ramsey theory Categories:46B15, 05D10 |
3. CMB 2008 (vol 51 pp. 413)
Big Ramsey Degrees and Divisibility in Classes of Ultrametric Spaces Given a countable set $S$ of positive reals, we study
finite-dimensional Ramsey-theoretic properties of the countable
ultrametric Urysohn space $\textbf{Q} _S$ with distances in $S$.
Keywords:Ramsey theory, Urysohn metric spaces, ultrametric spaces Categories:05C50, 54E35 |
4. CMB 2007 (vol 50 pp. 632)
Transformations and Colorings of Groups Let $G$ be a compact topological group and let $f\colon G\to G$ be a
continuous transformation of $G$. Define $f^*\colon G\to G$ by
$f^*(x)=f(x^{-1})x$ and let $\mu=\mu_G$ be Haar measure on $G$. Assume
that $H=\Imag f^*$ is a subgroup of $G$ and for every
measurable $C\subseteq H$,
$\mu_G((f^*)^{-1}(C))=\mu_H(C)$. Then for every measurable
$C\subseteq G$, there exist $S\subseteq C$ and $g\in G$ such that
$f(Sg^{-1})\subseteq Cg^{-1}$ and $\mu(S)\ge(\mu(C))^2$.
Keywords:compact topological group, continuous transformation, endomorphism, Ramsey theoryinversion, Categories:05D10, 20D60, 22A10 |