Two-colorings of a dense subgroup of \({\mathbb{Q}}^ n\) that forbid many distances
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.
scientific article; zbMATH DE number 4148099