On Beck's coloring of posets
The authors study Beck-like coloring of partially ordered sets with a least element 0. To any poset \(P\) with \(0\) they assign a graph, called a zero-divisor graph, whose vertices are labelled by the elements of \(P\) with two vertices \(x,y\) adjacent if \(0\) is the only element lying below \(x\) and \(y\). They prove that for such graphs the chromatic number and the clique number coincide. Further, they give a condition under which posets are not finitely colorable. The main result of the paper is similar to Dilworth's theorem.
scientific article; zbMATH DE number 5644978