Let G be a group, X be an infinite transitive G-space. A free ultrafilter U on X is called G-selective if, for any G-invariant partition P of X, either one cell of P is a member of U , or there is a member of U which meets each cell of P in at most one point. We show (Theorem 1) that in ZFC with no additional set-theoretical assumptions there exists a G-selective ultrafilter on X, describe all G-spaces X (Theorem 2) such that each free ultrafilter on X is G-selective, and prove (Theorem 3) that a free ultrafilter U on ω is selective if and only if U is G-selective with respect to the action of any countable group G of permutations of ω.A free ultrafilter U on X is called G-Ramsey if, for any G-invariant coloring χ : [G] 2 → {0, 1}, there is U ∈ U such that [U ] 2 is χ-monochrome. By Theorem 4, each G-Ramsey ultrafilter on X is Gselective. Theorems 5 and 6 give us a plenty of Z-selective ultrafilters on Z (as a regular Z-space) but not Z-Ramsey. We conjecture that each Z-Ramsey ultrafilter is selective.
AMS Classification: 05D10, 54H15Keywords: G-space, G-selective and G-Ramsey ultrafilters, the Stone-Čech compactification A free ultrafilter U on an infinite set X is said to be selective if, for any partition P of X, either one cell of P is a member of U, or some member of U meets each cell of P in at most one point. The selective ultrafilters on ω = {0, 1, . . .} are also known under the name Ramsey ultrafilters (see, for example [1]) because U is selective if and only if, for each coloring χ : [ω] 2 → {0, 1} of 2-element subsets of ω, there exists U ∈ U such that the restriction χ| [U ] 2 ≡ const.
A ballean is a set X endowed with some family F of its subsets, called the balls, in such a way that (X, F) can be considered as an asymptotic counterpart of a uniform topological space. Given a cardinal κ, we define F using a natural order structure on κ. We characterize balleans up to coarse equivalence, give the criterions of metrizability and cellularity, calculate the basic cardinal invariant of these balleans. We conclude the paper with discussion of some special ultrafilters on cardinal balleans.2010 MSC: 54A25; 05A18.
Let $G$ be a group and $X$ be a $G$-space with the action $G\times X\rightarrow X$, $(g,x)\mapsto gx$. A subset $F$ of $X$ is called a kaleidoscopical configuration if there exists a coloring $\chi:X\rightarrow C$ such that the restriction of $\chi$ on each subset $gF$, $g\in G$, is a bijection. We present a construction (called the splitting construction) of kaleidoscopical configurations in an arbitrary $G$-space, reduce the problem of characterization of kaleidoscopical configurations in a finite Abelian group $G$ to a factorization of $G$ into two subsets, and describe all kaleidoscopical configurations in isometrically homogeneous ultrametric spaces with finite distance scale. Also we construct $2^{\mathfrak c}$ (unsplittable) kaleidoscopical configurations of cardinality $\mathfrak c$ in the Euclidean space $\mathbb{R}^n$.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.