The Erdos Distance Problem}
Erdos posed the following question:
Let 
be the min number of distinct distances that you
can get with 
points in 
. Try to find asymptotically.
For example, if 
(the most studied case), this amounts to
two questions:
(1) How can you place points in the plane to minimize how many different
distinct distances there are, and
(2) Given any set of points in the plane, how many distinct distances
are you guaranteed.
Surveys and Writeups
by Garibaldi, Iosevich
Alex Iosevich (about the Szemeredi-Trotter thm
by William Gasarch. Unpublished
Papers that made Progress in Rough Chronological Order that used Elementary techniques
On sets of distances of n points by Erdos. American Math Monthly 53:248-250, 1946. 
On the different distances determined for n points by Moser. 
The number of different distances determined by n points in the plane by Chung. Journal of Combinatorial Theory Series A, 36:342–354, 1984. 
The number of different distances determined by a set of points in the Eucidean plane. by Chung, Szemeredi, Trotter. Discrete and Computational Geometry 7:1-11, 1992. 
Crossing numbers and hard Erdos problems in discrete geometry by Szekely. Combinatorics, Probability, and Computing 11:1-10, 1993. 
. This is a great paper that has a much easier proof than earlier papers.
Near optimal bounds for the Erdos distinct distances problem in high dimensions by Solymosi and Toth. Combinatorics 28:113-125, 2008. 
. A diff paper says: A close inspection of the general Solymosi-Toth approach shows that, without any additional geometric idea, it can never lead to a lower bound greater than 
. Note 8/9 is 0.888…
On distinct distance by G. Tardos. Advances in Mathematics, 180:275-289, 2003. 
. Exp is roughly 0.8635
Papers that made Progress in Rough Chronological Order that used Advanced Techniques
[KT.pdf A new entropy inequality for the Erdos distance problem by
N. Katz and G.Tardos. Contemporar Mathematics Volume 342.
. Exp is approx 0.8641]
in this paper; however, Guth and Katz use the Elekes-Sharir framework to obtain the result in the next item.
and 
.
for any 
. Exp is approx 0.546.
