Lower bounds on off-diagonal Ramsey numbers

The goal of this post is to prove the following elementary lower bound for off-diagonal Ramsey numbers (where is fixed and we are interested in the asymptotic behavior as gets large): The proof does not make use of the Lovász local lemma, which improves the bound by a factor of ; nevertheless, I think it’s [...]

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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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A little more about zeta functions and statistical mechanics

In the previous post we described the following result characterizing the zeta distribution. Theorem: Let be a probability distribution on . Suppose that the exponents in the prime factorization of are chosen independently and according to a geometric distribution, and further suppose that is monotonically decreasing. Then for some real . I have been thinking [...]

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Zeta functions, statistical mechanics and Haar measure

An interesting result that demonstrates, among other things, the ubiquity of in mathematics is that the probability that two random positive integers are relatively prime is . A more revealing way to write this number is , where is the Riemann zeta function. A few weeks ago this result came up on math.SE in the [...]

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Test your intuition: consecutive tails

Something very unfortunate has happened: several things I have recently written that could have been blog entries are instead answers on math.SE! In the interest of exposition beyond the Q&A format I am going to “rescue” one of these answers. It is an answer to the following question, which I would like you to test [...]

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