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

Let { p 1 , … , p n } ⊆ R d . 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 is at least Ω ( ( n 1 / ( 6 d ) ( log n ) 1 / 3 ) / d 1 / 3 ) . This is not the best known; however the technique is new. 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 ( X 3 ) are different? If d = 2 then the the size is at least Ω ( ( log log n ) 1 / 186 ) ; if d = 3 then the size is at least Ω ( ( log log n ) 1 / 396 ) . our proofs use variants of the canonical Ramsey theorem and some geometric lemmas.