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Proceedings of the London Mathematical Society 2006 93(3):545-569; doi:10.1017/S0024611506015887
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© London Mathematical Society

Sparse Partition Regularity

Imre Leader and Paul A. Russell

Department of Pure Mathematics and Mathematical Statistics, Centre for Mathematical Sciences Wilberforce Road, Cambridge, CB3 0WB, United Kingdom I.Leader{at}dpmms.cam.ac.uk, P.A.Russell{at}dpmms.cam.ac.uk

Received 3 June 2005. Revision received 25 October 2005.

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 mp, 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–Rödl 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. 2000 Mathematics Subject Classification 05D10.


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