A New application of Ramsey Theory to a Geometry problem

All of the material summazized here is in a new paper by Gasarch and Zbarsky. You can find that paper here)

Consider the following problem:

1. Let {p1,...,pn} be a subset of distinct points in R^d. We think of d ≤ n. How big is the largest subset X of points such that all of the distances determined by pairs of elements of X are different? Let h(d,n) be the min size of X. That is, we always have a set X of size h(d,n) with all distances between points different.
   
2. Assume that no three of the original points are collinear. How big is the largest subset X of points such that all of the areas determined by triples of points in X are different? Let g(d,n) be the min size of X. That is, we always have a set X of size g(d,n) with all triangle-areas of different sizes.
   

(This is NOT the Erdos Distance problem where you are given n points and want to know the min number of diff distances you have.) What is know about the problem posed?

1. The case of d=1 is known h(1,n)=Θ(sqrt(n)).
   
2. Erdos said somewhat mysterioulsy (I paraphrase)
   

   It is easy to see that there are constants ep(d) such that h(d) ≥ n^ep(d). (BILL: Frankly- I do not find it easy to see.)

   Aside from that, not much was known.
   

Until now. Gasarch and Zbarsky have shown the following (roughly)

1. h(d,n) ≥ Ω(n^1/(6d)).
   
2. g(2,n) ≥ Ω((log log n)^1/2901).
   
3. g(3,n) ≥ Ω((log log n)^1/27804).
   

We believe these results are new. They were obtained using variants of the Canonical Ramsey Theorem, originally due to Erdos and Rado, and some geometric lemmas.