Yet another species of forbidden-distances chromatic number (Q2721659)

The upper chromatic number of a metric space \((X,d)\) is the smallest positive integer \(m\) such that for each sequence \(s_1,\dots, s_m> 0\) there is a partition of \(X\) into sets \(X_1,\dots, X_m\) (here allowed to be empty) such that no two points of \(X_i\) are a distance \(s_i\) apart, \(i= 1,\dots, m\). The authors promote the study of another sequence of forbidden-distance chromatic numbers related to the upper chromatic numbers, but seemingly much more tractable: for \(k\in\mathbb{N}\) and metric space \((X,d)\), \(B_k(X)\) is the smallest \(m\in\mathbb{N}\) (if any) such that for every set of \(k\) or fewer positive numbers, there is a coloring of \(X\) with \(m\) colors with every distance in the set forbidden for every color.