Let \(\nu(\mathbb{Z}^m)\) be the minimal number of colors enough to color the \(m\)-dimensional integer grid \(\mathbb{Z}^m\) so that there would be no infinite monochromatic symmetric subsets. Banakh and Protasov [3] compute \(\nu(\mathbb{Z}^m) = m+1\). For the one-dimensional case this just means that one can color positive integers in red, while negative integers in blue, thereby avoiding an infinite monochromatic symmetric subset by a trivial reason. This motivates the question what changes if we allow only colorings unlimited in both directions (in “all” directions for \(m > 1\)). In this paper we show that then \(\nu(\mathbb{Z})\) increases by \(1\), whereas for higher dimensions the values \(\nu(\mathbb{Z}^m)\) remain unaffected.
Furthermore we examine the density properties of a set \(A \subseteq \mathbb{Z}^m\) that ensure the existence of infinite symmetric subsets or arbitrarily large finite symmetric subsets in \(A\). In the case that \(A\) is a sequence with small gaps, we prove a multi-dimensional analogue of the Szemerédi theorem, with symmetric subsets in place of arithmetic progressions. A similar two-dimensional statement is known for collinear subsets (Pomerance [10]), whereas for two-dimensional arithmetic progressions even the corresponding version of van der Waerden’s theorem is known to be false.
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