Unit-distance graphs, graphs on the integer lattice and a Ramsey type result
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Let (R2, 1) denote the graph withR2 as the vertex set and two vertices adjacent if and only if their Euclidean distance is 1. The problem of determining the chromatic numberχ(R2, 1) is still open; however,χ(R2, 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 (R2, 1). By a recent theorem of Chilakamarri, it is enough to consider certain graphs on the integer lattice. More precisely, forr > 0, let (Z2,r,\(\sqrt 2 \)) denote a graph with vertex setZ2 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 inZ2 if there exists a real numberd > 0 such that, for arbitrarily larger, G is isomorphic to a subgraph of (Z2,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 (R2, 1) if and only ifG is faithfully\(\sqrt 2 \)-recurring inZ2. In this paper we prove thatχ(Z2,r,\(\sqrt 2 \)) ≥ 5 for integersr ≥ 1. We also prove a Ramsey type result which states that for any integerr > 1, and any coloring ofZ2 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.
AMS (1991) subject classification05C15 05C55
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