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Search: MSC category 05D10 ( Ramsey theory [See also 05C55] )

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1. CMB 2012 (vol 56 pp. 709)

Bartošová, Dana
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 Fraïssé 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)

Service, Robert
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 2007 (vol 50 pp. 632)

Zelenyuk, Yevhen; Zelenyuk, Yuliya
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

4. CMB 1999 (vol 42 pp. 25)

Brown, Tom C.; Graham, Ronald L.; Landman, Bruce M.
On the Set of Common Differences in van der Waerden's Theorem on Arithmetic Progressions
Analogues of van der Waerden's theorem on arithmetic progressions are considered where the family of all arithmetic progressions, $\AP$, is replaced by some subfamily of $\AP$. Specifically, we want to know for which sets $A$, of positive integers, the following statement holds: for all positive integers $r$ and $k$, there exists a positive integer $n= w'(k,r)$ such that for every $r$-coloring of $[1,n]$ there exists a monochromatic $k$-term arithmetic progression whose common difference belongs to $A$. We will call any subset of the positive integers that has the above property {\em large}. A set having this property for a specific fixed $r$ will be called {\em $r$-large}. We give some necessary conditions for a set to be large, including the fact that every large set must contain an infinite number of multiples of each positive integer. Also, no large set $\{a_{n}: n=1,2,\dots\}$ can have $\liminf\limits_{n \rightarrow \infty} \frac{a_{n+1}}{a_{n}} > 1$. Sufficient conditions for a set to be large are also given. We show that any set containing $n$-cubes for arbitrarily large $n$, is a large set. Results involving the connection between the notions of ``large'' and ``2-large'' are given. Several open questions and a conjecture are presented.

Categories:11B25, 05D10

5. CMB 1997 (vol 40 pp. 149)

Brown, Tom C.; Landman, Bruce M.; Mishna, Marni
Monochromatic homothetic copies\\ of $\{1,1+s,1+s+t\}$
For positive integers $s$ and $t$, let $f(s, t)$ denote the smallest positive integer $N$ such that every $2$-colouring of $[1,N]=\{1,2, \ldots , N\}$ has a monochromatic homothetic copy of $\{1, 1+s, 1+s+t\}$. We show that $f(s, t) = 4(s+t) + 1$ whenever $s/g$ and $t/g$ are not congruent to $0$ (modulo $4$), where $g=\gcd(s,t)$. This can be viewed as a generalization of part of van~der~Waerden's theorem on arithmetic progressions, since the $3$-term arithmetic progressions are the homothetic copies of $\{1, 1+1, 1+1+1\}$. We also show that $f(s, t) = 4(s+t) + 1$ in many other cases (for example, whenever $s > 2t > 2$ and $t$ does not divide $s$), and that $f(s, t) \le 4(s+t) + 1$ for all $s$, $t$. Thus the set of homothetic copies of $\{1, 1+s, 1+s+t\}$ is a set of triples with a particularly simple Ramsey function (at least for the case of two colours), and one wonders what other ``natural'' sets of triples, quadruples, {\it etc.}, have simple (or easily estimated) Ramsey functions.

Category:05D10

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