Euclidean Szlam Numbers

Abstract

Suppose X ⊆ R n , for some positive integer n, is closed under vector addition, ρ a translation invariant distance function on X, and D ⊆ (0, ∞). The distance graph G ρ (R n , D) is the graph with vertex set R n with u, v ∈ R n adjacent if and only if ρ(u, v) ∈ D. A rather red coloring of G is a coloring of X with red and blue such that no two points adjacent in G are both blue. The Szlam number of G is the minimum cardinality, over all rather red colorings of G, of F ⊆ X such that no translate of F is all red. We exploit results of Johnson, Szlam, and Kloeckner to show that for every positive integer n there exists ρ such that the Szlam number of G ρ (R 2 , {1}) is n. Let K, X ⊆ R n for some positive integer n and suppose that K = −K and (0, . . . , 0) ∈ K. We will call such a set a K-set. Define G(X, K), a K-graph on X, to be the graph whose vertex set is X and x, y ∈ X are adjacent if and only if x − y ∈ K. We show that for every K-set there exists a translation invariant distance function on R n such that G(R n , K) = G ρ (R n , {1}) and for every translation invariant distance function ρ on R n and D ⊆ (0, ∞) there exists a K-set such that G(R n , K) = G ρ (R n , D). For certain K-sets we find the Szlam number of G ρ (R n , D) where G ρ (R, D) = G(R n , K).