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Combinatorial Number Theory

Background. The integer distance graph G(Z, D) with distance set D = {d1, d2, . . .} has the set of integers Z as the vertex set and two vertices x, y ∈ Z are adjacent if and only if |x − y| ∈ D. The integer distance graphs (under Euclidean norm) were first systematically studied by Eggleton–Erdős–Skilton in 1985 [12, 13], and have been investigated in many ways [50, 56, 57, 61]. One of main goals in these problems is characterizing prescribed distance sets that make the corresponding distance graphs to have finite chromatic number. Ruzsa, Tuza, and Voigt [50] gave a sufficient condition for χ(G(Z, D)) to be finite: