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<p>We give two extensions of the recent theorem of the first author that the odd distance graph has unbounded chromatic number. The first is that for any non-constant polynomial <i>f</i> with integer coefficients and positive leading coefficient, every finite colouring of the plane contains a monochromatic pair of distinct points whose distance is equal to <i>f</i>(<i>n</i>) for some integer <i>n</i>. The second is that for every finite colouring of the plane, there is a monochromatic pair of points whose distance is a prime number.</p>

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Prime and polynomial distances in colourings of the plane

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年11月30日

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  • James Davies,
  • Rose McCarty,
  • Michał Pilipczuk

摘要

We give two extensions of the recent theorem of the first author that the odd distance graph has unbounded chromatic number. The first is that for any non-constant polynomial f with integer coefficients and positive leading coefficient, every finite colouring of the plane contains a monochromatic pair of distinct points whose distance is equal to f(n) for some integer n. The second is that for every finite colouring of the plane, there is a monochromatic pair of points whose distance is a prime number.