A Higher Dimensional Version of a Problem of Erdős

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 ( X 2 ) are different? We show that |X| ≥ Ω(n 1 3d−3+o(1) ). This improves on the best known result which was |X| ≥ Ω(n 1 3d−2 ). Assume that no a of the points are on the same (a− 1)-hyperplane. How big is the largest subset X of points such that all of the volumes determined by elements of ( X a ) are different? We show that |X| ≥ Ω(n 1 (2a−1)d ). This concept had not been studied before. Let α be a regular cardinal between א0 and 2א0 . Let X ⊆ Rd such that no a of the original points are in the same (a − 1)-hyperplane. We show that there is an α-sized subset of X such that all of the volumes determined by elements of ( X a ) are different. We give two proofs: one assuming the Axiom of Choice and one assuming the Axiom of Determinacy.