Two-colorings of a dense subgroup of \({\mathbb{Q}}^ n\) that forbid many distances
type: Item
instance of: scholarly article
title: Two-colorings of a dense subgroup of \({\mathbb{Q}}^ n\) that forbid many distances
zbMATH Open document ID: 0701.05005
DOI: 10.1016/0012-365X(90)90033-E
author: P. D. jun. Johnson
published in: Discrete Mathematics
publication date: 1990-01-01T00:00:00Z
review text: Suppose G is an Abelian group and \(S\subset G\). A colouring (partition) of G forbids S if and only if, for each \(g\in G,\quad s\in S,\) g and \(g+s\) receive different colours (belong to different sets of the partition). Suppose (X,d) is a Euclidean space, and D is a set of positive numbers. A colouring of X forbids the distances D if and only if, for any x,y\(\in X\), if d(x,y)\(\in D\) then x and y receive different colours. Suppose n is a positive integer, \(A_ n\) is the subgroup of \(Q^ n\) consisting of rational points with odd denominator, and \(K=\{\sqrt{p/q};\) p and q are odd positive integers\(\}\). The author proves that for any positive integer n, there is a two- colouring of \(A_ n\) which forbids the distances K. From this, the author also deduces that there are a two-colouring of \(Q^ 3\), and a four-colouring of \(Q^ 4\), which forbid the distances K. These results improve some known results.
Mathematics Subject Classification ID: 05A17
Mathematics Subject Classification ID: 20K15
zbMATH DE Number: 4148099
zbMATH Keywords: Abelian group
zbMATH Keywords: partition
zbMATH Keywords: two-colouring of \(Q^ 3\)
zbMATH Keywords: four-colouring of \(Q^ 4\)
reviewed by: H. P. Yap
MaRDI profile type: Publication
cites work: Q4187840
cites work: Q4939321
cites work: Coloring abelian groups
cites work: Distances forbidden by some two-coloring of \(\mathbb{Q}^ 2\)
full work available at URL: 0012-365x(90)90033-e
OpenAlex ID: W2059169626
Wikidata QID: Q127613808
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description: scientific article; zbMATH DE number 4148099