Let $G$ be a group, $X$ be an infinite transitive $G$-space. A freeultrafilter $\UU$ on $X$ is called $G$-selective if, for any $G$-invariantpartition $\PP$ of $X$, either one cell of $\PP$ is a member of $\UU$, or thereis a member of $\UU$ which meets each cell of $\PP$ in at most one point. Weshow (Theorem 1) that in ZFC with no additional set-theoretical assumptionsthere 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 $\UU$ on $\omega$ is selective if and onlyif $\UU$ is $G$-selective with respect to the action of any countable group $G$of permutations of $\omega$. A free ultrafilter $\UU$ on $X$ is called $G$-Ramsey if, for any$G$-invariant coloring $\chi:[G]^2 \to \{0,1\}$, there is $U\in \UU$ such that$[U]^2$ is $\chi$-monochrome. By Theorem 4, each $G$-Ramsey ultrafilter on $X$is $G$-selective. Theorems 5 and 6 give us a plenty of $\mathbb{Z}$-selectiveultrafilters on $\mathbb{Z}$ (as a regular $\mathbb{Z}$-space) but not$\mathbb{Z}$-Ramsey. We conjecture that each $\mathbb{Z}$-Ramsey ultrafilter isselective.
展开▼
机译：假设$ G $为组，$ X $为无限可传递$ G $空间。如果对于$ X $的任何$ G $ -invariantpartition $ \ PP $，$ \ PP $的一个单元格都是$ \ UU的成员，则$ X $上的freeultrafilter $ \ UU $称为$ G $ -selective。 $，或者$ \ UU $的成员最多在一个点与$ \ PP $的每个单元格相遇。我们显示（定理1），在ZFC中没有附加的集理论假设，其中在$ X $上存在$ G $选择性超滤器，描述所有$ G $空间$ X $（定理2），使得$ X上的每个自由超滤器$是$ G $选择性的，并证明（定理3）当且仅当$ \ UU $对于任何可数的作用是$ G $选择性时，在$ \ omega $上的自由超滤器$ \ UU $才是选择性的$ \ omega $的排列中的$ G $组。如果对于任何$ G $不变色$ \ chi：[G] ^ 2 \ to \ {0,1 \} $，有一个在$ X $上的免费超滤器$ \ UU $被称为$ G $ -Ramsey。 $ U \ in \ UU $，使得$ [U] ^ 2 $是$ \ chi $单色。根据定理4，在$ X $上的每个$ G $ -Ramsey超滤器都是$ G $选择性的。定理5和定理6在$ \ mathbb {Z} $（作为常规$ \ mathbb {Z} $-空间）上为我们提供了很多$ \ mathbb {Z} $-selectiveultrafilters，但没有$ \ mathbb {Z} $-拉姆西。我们推测每个$ \ mathbb {Z} $-Ramsey超滤器都是选择性的。
展开▼
