Many problems in discrete geometry and additive combinatorics ask for the fewest distinct equivalence classes that may be determined by a set of n points or numbers under some algebraically defined equivalence relation. For example, Erdős asked for the fewest distinct distances determined by any set of n points in the Euclidean plane.
I will give a brief survey of some classic and recent results on questions of this sort. I will then discuss a recent result on the fewest distinct perpendicular bisectors determined by any set of n points in the plane.