It is known that 4 ≤ x(ℝ2) ≤ 7, where x(ℝ2) is the number of colour necessary to colour each point of Euclidean 2-space so that no two points lying distance 1 apart have the same colour. Any lattice-sublattice colouring sucheme for R2 must use at least 7 colour to have an excluded distance. This article shows that at least 6 colours are necessary for an excluded distance when convex polygonal tiles (all with area greater than some positive constant) are used as the colouring base.
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