OAI.EuclideanRamsey.classification

The theorem states that, for natural numbers s ≥ 2 and d ≥ 1 and an injective configuration a of s points in d-dimensional Euclidean space whose affine span over ℝ is all of ℝ^d, the configuration is Ramsey if and only if it satisfies the field criterion. Ramsey means that for every number of colours r ≥ 2 there is a dimension D ≥ 1 such that every colouring c of ℝ^D with r colours admits a monochromatic congruent copy of a, that is, points b₀,...,b_{s-1} in ℝ^D with |bᵢ−bⱼ| = |aᵢ−aⱼ| for all i, j, all given the same colour. The field criterion is formulated over the field K generated over ℚ by all coordinates of the points aᵢ, inside ℝ. It asks for a matrix P, indexed by {none, 1,...,d} (the augmented index set), with entries in the tensor product K ⊗_ℚ K, such that two conditions hold. First, for every point i, using the augmented vector (1, aᵢ₁,...,aᵢd), the sum over α, β of (augmented(i,α) ⊗ 1) · P(α,β) · (1 ⊗ augmented(i,β)) equals 0. Second, applying the multiplication map K ⊗_ℚ K → K to each entry P(α,β) with α and β both coordinate indices gives 1 when α = β and 0 otherwise.