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FORBIDDEN DISTANCES IN THE RATIONALS AND THE REALS

Published online by Cambridge University Press:  24 April 2006

NEIL HINDMAN ,
IMRE LEADER  and
DONA STRAUSS
NEIL HINDMAN
Affiliation:
Department of Mathematics, Howard University, Washington, DC 20059, USAnhindman@aol.comhttp://members.aol.com/nhindman/
IMRE LEADER
Affiliation:
Department of Pure Mathematics and Mathematical Statistics, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WB, United Kingdomi.leader@dpmms.cam.ac.uk
DONA STRAUSS
Affiliation:
Department of Pure Mathematics, University of Hull, Hull HU6 7RX, United Kingdomd.strauss@hull.ac.uk
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Abstract

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 $\{ c_n:n<\omega\}$ 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 $c_nx$. This result is motivated by some questions in partition regularity.

Keywords

05D10 (primary)28A05 (secondary)
Type
Notes and Papers
Information
Journal of the London Mathematical Society , Volume 73 , Issue 2 , April 2006 , pp. 273 - 286
Copyright
The London Mathematical Society 2006

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