Unit-distance graphs, graphs on the integer lattice and a Ramsey type result

Abstract

Let (R 2, 1) denote the graph withR 2 as the vertex set and two vertices adjacent if and only if their Euclidean distance is 1. The problem of determining the chromatic numberχ(R 2, 1) is still open; however,χ(R 2, 1) is known to be between 4 and 7. By a theorem of de Bruijn and Erdos, it is enough to consider only finite subgraphs of (R 2, 1). By a recent theorem of Chilakamarri, it is enough to consider certain graphs on the integer lattice. More precisely, forr > 0, let (Z 2,r, $$\sqrt 2 $$ ) denote a graph with vertex setZ 2 and two vertices adjacent if and only if their Euclidean distance is in the closed interval [r − $$\sqrt 2 $$ ,r + $$\sqrt 2 $$ ]. A simple graph is faithfully $$\sqrt 2 $$ -recurring inZ 2 if there exists a real numberd > 0 such that, for arbitrarily larger, G is isomorphic to a subgraph of (Z 2,r, $$\sqrt 2 $$ ) in which every pair of vertices are at least distancedr apart. Chilakamarri has shown that, ifG is a finite simple graph, thenG is isomorphic to a subgraph of (R 2, 1) if and only ifG is faithfully $$\sqrt 2 $$ -recurring inZ 2. In this paper we prove thatχ(Z 2,r, $$\sqrt 2 $$ ) ≥ 5 for integersr ≥ 1. We also prove a Ramsey type result which states that for any integerr > 1, and any coloring ofZ 2 either there exists a monochromatic pair of vertices with their distance in the closed interval [r − $$\sqrt 2 $$ ,r + $$\sqrt 2 $$ ] or there exists a set of three vertices closest to each other with three distinct colors.