Premium
Forbidden Distances in the Rationals and the Reals
Author(s) -
Hindman Neil,
Leader Imre,
Strauss Dona
Publication year - 2006
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610706022800
Subject(s) - rational number , partition (number theory) , mathematics , real number , natural number , combinatorics , class (philosophy) , discrete mathematics , set (abstract data type) , computer science , artificial intelligence , programming language
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 < 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 c n x . This result is motivated by some questions in partition regularity.