×

On asymmetric colorings of integer grids. (English) Zbl 1073.05068

A set \(A\subset {\mathbb Z}^m\) is called asymmetric if it contains no infinite set \(B\subset A\) such that \(B=g-B\) for some \(g\in {\mathbb Z}^m\); such \(B\) is called symmetric. A coloring of \(A\subset {\mathbb Z}^m\) is asymmetric if each color class is asymmetric. We mention some of the results in the article under review. There is an asymmetric 3-coloring of \({\mathbb Z}^2\) such that each ray passing through two lattice points intersects every color class. If \(\limsup _{n\to \infty }| A\cap [n]| /\sqrt {n}=\infty \) then \(A\subset {\mathbb N}\) contains arbitrarily large symmetric subsets; there is \(A\) for which this limsup is positive and finite and \(A\) contains no symmetric subset with more than \(2\) elements. If \(A=\{a_1,a_2,\dots \}\subset {\mathbb Z}^m\) is such that \(\liminf _{k\to \infty }{1\over k}\sum _{i=1}^k\| a_{i+1}-a_i\| <\infty \), where \(\| *\| \) is the sup-norm, then \(A\) contains arbitrarily large symmetric subsets. There is a set \(A\subset {\mathbb Z}^2\) such that \(\sum _{a\in A}1/\| a\| \) diverges but \(A\) contains no symmetric subset of cardinality more than \(4\).

MSC:

05D10 Ramsey theory
PDFBibTeX XMLCite