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Scale the Cairo pentagonal tiling so the short side is of length 1. Then it is easy to colour the tiling with 8 colours, two parallel ribbons of four colours each, to establish that the chromatic number of the plane is at most 8. We know that the chromatic number of the plane is at most 7, and at least 5 (Aubrey de Grey).

Attempting a colouring with 6 colours quickly leads to obstacles. What about colouring with 7 colours? There are many constraints, but so far I haven't found an obstacle.

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  • $\begingroup$ Are you asking for a coloring of the plane with seven colors? Or are you asking for a coloring of the Cairo tiling with seven colors? $\endgroup$
    – Gerry Myerson
    Commented Jun 18, 2018 at 1:07
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    $\begingroup$ Note the following comment taken from the new Polymath wiki regarding the chromatic number of the plane: A "tile-based" colouring of the plane must have at least 6 colours, as shown by Townsend [Tow2005]; [...] In [T1999], Thomassen showed that any tiling-based 6-coloring would have to be be "unscaleable", i.e. the maximum diameter of a tile and the minimum separation of same-coloured tiles must both be exactly 1 (so that the distance 1 is excluded by suitable colouring of the boundary points). $\endgroup$
    – Dan Rust
    Commented Jun 18, 2018 at 12:20
  • $\begingroup$ I know that a tiling with regular hexagons can be used to prove that the chromatic number of the plane is at most seven. I'm wondering whether the Cairo pentagonal tiling can manage this. I don't have this automated, but I do seem to be able to grow the coloring without conflicts. I will soon need something larger than a letter sized sheet of paper. $\endgroup$
    – Michael Ruxton
    Commented Jun 18, 2018 at 18:12

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