Applications of the Canonical Ramsey Theorem to Geometry

Let P be a set of n points in R^d. How big is the largest subset X of P such that all of the distances determined between pairs are different? We show that X is at at least Omega(n^{1/6d}) This is not the best known; however the technique is new. Assume that no three of the original points are collinear. How big is the largest subset X of P such that all of the areas determined by elements of all triples are different? We show that, if d=2 then X is at least Omega((log log n)^{1/186}) and if d=3 then X is at least Omega((log log n)^{1/396}). We also obtain results for countable sets of points in R^d. All of our proofs use variants of the canonical Ramsey theorem and some geometric lemmas.