Author Archives: apgoucher

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 →

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 →

(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 →

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 →

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 →

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 →

Many theorems in analysis have combinatorial analogues, which turn out to be equivalent (in the sense that one can be derived from the other, and vice-versa, using significantly less maths than is necessary to prove either from first principles). A particularly … Continue reading →

Yesterday, I attended the UKMT mentoring conference in Murray Edwards College (affectionately abbreviated to ‘Medwards’), which mainly consisted of an excellent dinner in the Fellows’ dining room and informal discussion about various topics. Speaking of mentoring, I recently prepared for one of … Continue reading →

A few months ago, Yitang Zhang announced that there are infinitely many pairs of primes, separated by a distance no more than 70000000. This initiated an extensive collaborative effort (known as polymath8) to reduce this bound by optimising different parts of the proof, until … Continue reading →

(There appears to be a recent week-long awkward silence on cp4space, partially due to the end of Season II of ciphers. Here’s an attempt to rectify it.) Suppose we have a square drum, consisting of a membrane fixed at its perimeter. The membrane … Continue reading →