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Journal of the London Mathematical Society 2006 73(2):273-286; doi:10.1112/S0024610706022800
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© The London Mathematical Society

Forbidden Distances in the Rationals and the Reals

Neil Hindman, Imre Leader and Dona Strauss

Department of Mathematics, Howard University Washington, DC 20059, USA nhindman{at}aol.com http://members.aol.com/nhindman/
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
Department of Pure Mathematics, University of Hull Hull HU6 7RX, United Kingdom d.strauss{at}hull.ac.uk

Received 14 May 2003. Revision received 7 June 2005.

Our main aim in this paper is to show that there is a partition of the reals into finitely many classes with ‘many’ forbidden distances, in the following sense: for each positive real x, there is a natural number n such that no two points in the same class are at distance x/n. In fact, more generally, given any infinite set {cn:n < {omega}} of positive rationals, there is a partition of the reals into three classes such that for each positive real x, there is some n such that no two points in the same class are at distance cnx. This result is motivated by some questions in partition regularity.


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