Simple Search

Query:

Enter a query and click »Search«...

Format:

Display: entries per page entries

Zbl 0701.05005
Johnson, Peter D.jun.
Two-colorings of a dense subgroup of ${\bbfQ}\sp n$ that forbid many distances. (English)
[J] Discrete Math. 79, No.2, 191-195 (1990). ISSN 0012-365X

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). \par 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. \par Suppose n is a positive integer, $A\sb n$ is the subgroup of $Q\sp n$ consisting of rational points with odd denominator, and $K=\{\sqrt{p/q};$ p and q are odd positive integers$\}$. \par The author proves that for any positive integer n, there is a two- colouring of $A\sb n$ which forbids the distances K. From this, the author also deduces that there are a two-colouring of $Q\sp 3$, and a four-colouring of $Q\sp 4$, which forbid the distances K. These results improve some known results.
[H.P.Yap]

MSC 2000:: *05A17 Partitions of integres (combinatorics)
20K15 Torsion free abelian groups, finite rank

Keywords: Abelian group; partition; two-colouring of $Q\sp 3$; four-colouring of $Q\sp 4$

Cited in: Zbl 0771.05039

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Article Journal

	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.
	Elementary number theory. Primes, congruences, and secrets.

Simple Search

Zentralblatt MATH Berlin [Germany]