Applications of Ramsey's theorem

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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.