Applications of the Erdős-Rado Canonical Ramsey Theorem to Erdős-Type Problems

Let {p1,…,pn}⊆Rd. We think of d≪n. How big is the largest subset X of points such that all of the distances determined by elements of are different? We show that X is at least . This is not the best known; however the technique is new.Assume that no 3 of the original points are in the line. How big is the largest subset X of points such that all of the areas determined by elements of are different? If d=2 then the the size is at least ; if d=3 then the size is at least .All of our proofs use variants of the canonical Ramsey theorem and some geometric lemmas.