On list coloring with separation of the complete graph and set system intersections

Abstract

We consider the following list coloring with separation problem: Given a graph G and integers a, b, find the largest integer c such that for any list assignment L of G with |L(v)| = a for any vertex v and |L(u) ∩ L(v)| ≤ c for any edge uv of G, there exists an assignment φ of sets of integers to the vertices of G such that φ(u) ⊂ L(u) and |φ(v)| = b for any vertex u and φ(u) ∩ φ(v) = ∅ for any edge uv. Such a value of c is called the separation number of (G, a, b). Using a special partition of a set of lists for which we obtain an improved version of Poincaré’s crible, we determine the separation number of the complete graph Kn for some values of a, b and n, and prove bounds for the remaining values.