FORBIDDEN DISTANCES IN THE RATIONALS AND THE REALS

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 < ω} 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.