Monochromatic triangles in two-colored plane

Abstract

We prove that for any partition of the plane into a closed set C and an open set O and for any configuration T of three points, there is a translated and rotated copy of T contained in C or in O.

Apart from that, we consider partitions of the plane into two sets whose common boundary is a union of piecewise linear curves. We show that for any such partition and any configuration T which is a vertex set of a non-equilateral triangle there is a copy of T contained in the interior of one of the two partition classes. Furthermore, we characterize the “polygonal” partitions that avoid copies of a given equilateral triple.

These results support a conjecture of Erdős, Graham, Montgomery, Rothschild, Spencer and Straus, which states that every two-coloring of the plane contains a monochromatic copy of any nonequilateral triple of points; on the other hand, we disprove a stronger conjecture by the same authors, by providing non-trivial examples of two-colorings that avoid a given equilateral triple.

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Correspondence to Vít Jelínek.

Additional information

ITI is supported by project 1M0021620808 of the Czech Ministry of Education.

KAM is supported by project MSM0021620838 of the Czech Ministry of Education.

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Jelínek, V., Kynčl, J., Stolař, R. et al. Monochromatic triangles in two-colored plane. Combinatorica 29, 699–718 (2009). https://doi.org/10.1007/s00493-009-2291-y

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Mathematics Subject Classification (2000)

  • 05D10
  • 52C10