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I would like to know the minimum number k such that the plane R^2 can be coloured with k colors such that no colour contain all the possible distances. In other words, a colouring such that each color forbids a distance (not necessarily the same as it is the case for http://en.wikipedia.org/wiki/Hadwiger%E2%80%93Nelson_problem)

Has this problem been studied? It is easy to prove that 2 colors isn't enough, but I don't know for 3 colors... We also know that this number k is at most 7 (using http://en.wikipedia.org/wiki/File:Hadwiger-Nelson.svg for example), but I did not find any better bound yet.

Bests,

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Seems to be the polychromatic number of the plane.

According to my knowledge, the value is at least 4 (due to Raiskii) and at most 6 (due to Stechkin).

See Chap. 4 and 6 of The Mathematical Coloring book.

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