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

Summary

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 Erdös, 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.