A new connection between two kinds of Euclidean coloring problems (Q2719866)

Let \(S\subseteq \mathbb{R}^n\) be given, together with a metrix \(\rho\) on \(\mathbb{R}^n\) and a set \(D\subseteq (0,\infty)\) of distances. One object of study is \(\chi_\rho(S,D)\), the smallest number of colors necessary to color the points of \(S\) so that no two of them a distance \(d\in D\) apart are colored alike. Now suppose that there are only two colors, red and blue. The question is: among red-blue colorings of \(S\) such that every \(d\in D\) is forbidden for blue, what ``configurations'' are forbidden for red? \textit{R. Juhasz} [J. Comb. Theory, Ser. A 27, 152-160 (1979; Zbl 0431.05001)] showed that if \(D= \{1\}\), the red set must contain congruent copies of every four-point planar set. \textit{A. D. Szlam} [J. Comb. Theory, Ser. A 93, 173-174 (2001; Zbl 0966.05077)] found a simple but powerful connection between such questions (but involving forbidden translates rather than congruent copies of configurations for red) and determining \(\chi_\rho(S, D)\), for various translation-invariant metrics \(\rho\). The present paper generalizes that connection.