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A (finite) matrix A over the positive integers is said to be partition regular (PR) if, whenever the positive integers are finitely coloured, there exists a vector x with all entries the same colour and satisfying Ax = 0. Many of the classical results of Ramsey Theory can be thought of as the statement that a certain matrix is PR.

In 1933, Rado provided a complete characterization of all PR matrices. In a refinement of Rado’s theorem, Deuber showed that a matrix A is PR if and only if solutions to Ax = 0 can always be found within a certain sort of ‘iterated arithmetic progression’ which he called an (m, p, c)-set. He further conjectured that one should be able to find a set S with the property that, whenever S is finitely coloured, there is a monochromatic (m, p, c)-set, but such that the collection of (m, p, c)-sets in S is very non-dense (in the sense that they form a hypergraph of large girth). Our proof of this conjecture is too long to give in full in this talk. Instead, I shall present some background material on sparse Ramsey theory and attempt to explain the two main notions on which the proof is based: an extension of Nesetril-Rodl amalgamation, and a Ramsey theorem for trees proved by Bergelson, Hindman and Leader.

This is joint work with Imre Leader.
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