Our aim in this paper is to prove Deuber's conjecture on sparse partition regularity, that for every m, p and c there exists a subset of the natural numbers whose (m,p,c)-sets have high girth and chromatic number. More precisely, we show that for any m, p, c, k and g there is a subset S of the natural numbers that is sufficiently rich in (m,p,c)-sets that whenever S is k-coloured there is a monochromatic (m,p,c)-set, yet is so sparse that its (m,p,c)-sets do not form any cycles of length less than g.
Our main tools are some extensions of Nesetril-Rodl amalgamation and a Ramsey theorem of Bergelson, Hindman and Leader. As a sideline, we obtain a Ramsey theorem for products of trees that may be of independent interest.
Paper (pdf)