Applications of Ramsey's theorem

关于我们研究的两个目的，我们得到了以下结果。(1) kappa-many棘序列扇是由kappa-many收敛序列的不相交并通过将所有极限点识别为单个点而得到的拓扑空间。本文的研究目的之一是利用Ramsey型组合学研究Kodama关于顺序风扇sigma积正态性的问题。Tamano和Feng证明了序列扇的可数积的子空间（更一般地说，Lasnev空间）当且仅当它具有可数扇紧度时是可度量的。Lasnev空间是一类广义度量空间。我们通过证明具有非mu空间的可数网络的空间的存在性和讨论sigma空间的各种定义，对广义度量空间的理论做出了贡献。(2)图在R^3中的嵌入称为图的空间嵌入。已知一个足够大的完备图总是包含一个不可分连杆和一个非平凡结，这是一种拉姆齐型定理。Negami证明了这样的定理适用于一种被称为良好绘图的嵌入。

We got the following results concerning two purposes of our research.(1) The sequential fan with kappa-many spines is the topological space obtained from the disjoint union of kappa-many convergent sequences by identifying all the limit point to a single point. One of the purposes of our research was to study Kodama's question on the normality of SIGMA-product of sequential fans by using Ramsey type combinatorics. Tamano and Feng proved that a subspace of a countable product of sequential fans (more generally, Lasnev spaces) is metrizable if and only if it has countable fan-tightness. The class of Lasnev spaces is a kind of generalized metric spaces. We contributed to the theory of generalized metric spaces by showing the existence of a space with a countable network which is not a mu-space and by discussing various definitions of SIGMA-spaces.(2) An embedding of a graph into R^3 is called a spatial embedding of the graph. It had been known that a sufficently large complete graph always include a nonsplittable link and a nontrivial knot, which is a kind of Ramsey type theorem. Negami proved that such a theorem holds for an embedding called good drawing.