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FORBIDDEN DISTANCES IN THE RATIONALS AND THE REALS
Published online by Cambridge University Press: 24 April 2006
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.
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- The London Mathematical Society 2006
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