Monthly Archives: December 2013

New Year’s Eve

It’s now New Year’s Eve, which is why the cp4space banner has changed again. As is usual for this time of year, there is a joint Anglo-Hungarian IMO training camp in Tata, Hungary. I’ve mentioned it before at the end of 2012, … Continue reading

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Applications of ultrafilters

This is the sequel to the previous article defining ultrafilters and establishing basic properties thereof. Again, the proof of Hindman’s theorem is from Imre Leader’s lecture course on Ramsey theory. It seems initially paradoxical that ultrafilters are actually useful. After … Continue reading

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Comprehending ultrafilters

This is the first of a projected two-part series of articles about ultrafilters. The former introduces ultrafilters and establishes various properties about them, so that the latter article can concentrate entirely on applications of ultrafilters. Much of this material is from Professor Imre … Continue reading

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Large cardinals

(This article is about the axioms concerned with the existence of enormous sets. If you were instead looking for corpulent subordinates of the Pope, then I suggest watching this Monty Python sketch.) How big is the universe? Ultrafinitists such as Doron … Continue reading

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Tarski’s circle-squaring problem

Is it possible to dissect a square into finitely many pieces, which can then be rearranged to form a disc of equal area? Note that, unlike in three dimensions where the Banach-Tarski paradox exists, dissections must respect the two-dimensional Banach measure … Continue reading

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Free commutative monoid on countably many generators

Consider the following problem: Let  be a finite set of positive integers. Suppose we colour the positive integers with k distinct colours. Prove that there exist constants a and b such that is monochromatic. Initially, this seems as though it … Continue reading

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Miscellaneous news and sphere eversion

Before we begin, there are a few late items of news worth discussing: There is a free edition of Wolfram Mathematica for the Raspberry Pi, an increasingly popular versatile low-cost computer. More details are available over at The Aperiodical. The … Continue reading

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