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

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## Summary

Let (**R**^{2}, 1) denote the graph with**R**^{2} as the vertex set and two vertices adjacent if and only if their Euclidean distance is 1. The problem of determining the chromatic number*χ*(**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, for*r* > 0, let (**Z**^{2},*r*,\(\sqrt 2 \)) denote a graph with vertex set**Z**^{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 in**Z**^{2} if there exists a real number*d* > 0 such that, for arbitrarily large*r, G* is isomorphic to a subgraph of (**Z**^{2},*r*,\(\sqrt 2 \)) in which every pair of vertices are at least distance*dr* apart. Chilakamarri has shown that, if*G* is a finite simple graph, then*G* is isomorphic to a subgraph of (**R**^{2}, 1) if and only if*G* is faithfully\(\sqrt 2 \)-recurring in**Z**^{2}. In this paper we prove that*χ*(**Z**^{2},*r*,\(\sqrt 2 \)) ≥ 5 for integers*r* ≥ 1. We also prove a Ramsey type result which states that for any integer*r* > 1, and any coloring of**Z**^{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.

## AMS (1991) subject classification

05C15 05C55## Preview

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## References

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