Suppose D ∈ (0, ∞) and 0 < |D| < ∞. The distance graph G(R, D) is the graph with vertex set R, and two vertices x, y are adjacent if |x — y| ∈D. We prove that for every positive integer t > 1 there is a distance set D such that the chromatic number of G(R, D) is t and no proper coloring of G(R, D) with t colors allows monochromatic intervals. This result disproves a conjecture in [2].
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机译：假设D∈（0，∞）并且0 <| D | <∞。距离图G（R，D）是顶点集为R的图，如果| x — y |，则两个顶点x，y相邻。 ∈D。我们证明，对于每个正整数t> 1，都有一个距离集合D，使得G（R，D）的色数为t，并且没有t色的G（R，D）正确着色允许单色间隔。这个结果证明了[2]中的一个猜想。
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