Abstract: In 1946, in a paper published in the American Mathematical Monthly, Erdos raised the following innocent looking question: At most how many times can a given distance occur among n points in the plane?

This and many similar geometric and number-theoretic questions have motivated a lot of research in extremal graph and hypergraph theory and have led to discovery of several important combinatorial tools. Nevertheless, we are still far from having a satisfactory answer to Erdos's original question. All known approaches to this problem are based on forbidden pattern arguments.

We outline some old and new ideas leading to the best known bound of O(n^{4/3}), and discuss whether such arguments can ever lead to the truth.