Ultrafilters in Ramsey theory

We continue our exploration of ultrafilters. Today we’ll discuss the infinite Ramsey theorem, which is the following classical result: Theorem: Suppose the complete graph on countably many vertices has its edges colored in one of colors. Then there is a monochromatic (i.e. an infinite subgraph all of whose edges are the same color). The finite [...]

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Ultrafilters in topology

Remark: To forestall empty set difficulties, whenever I talk about arbitrary sets in this post they will be non-empty. We continue our exploration of ultrafilters from the previous post. Recall that a (proper) filter on a poset is a non-empty subset such that For every , there is some such that . For every , [...]

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Boolean rings, ultrafilters, and Stone’s representation theorem

Recently, I have begun to appreciate the use of ultrafilters to clean up proofs in certain areas of mathematics. I’d like to talk a little about how this works, but first I’d like to give a hefty amount of motivation for the definition of an ultrafilter. Terence Tao has already written a great introduction to [...]

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