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 [...]
Posts Tagged ‘pigeonhole principle’
Ultrafilters in Ramsey theory
Posted in logic and set theory, Ramsey theory, tagged pigeonhole principle, ultrafilters on December 14, 2010 | 16 Comments »
Ramsey theory and Fermat’s Last Theorem
Posted in graph theory, number theory, Ramsey theory, tagged pigeonhole principle on October 13, 2010 | 3 Comments »
In the first few lectures of Graph Theory, the lecturer (Paul Russell) presented a cute application of Ramsey theory to Fermat’s Last Theorem. It makes a great introduction to the general process of casting a problem in one branch of mathematics as a problem in another and is the perfect size for a blog post, [...]
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