Abstract
It is shown that the number of distinct types of three-point hinges, defined by a real plane set of n points is ≫n2 log−3n, where a hinge is identified by fixing two pairwise distances in a point triple. This is achieved via strengthening (modulo a logn factor) of the Guth- Katz estimate for the number of pairwise intersections of lines in ℝ3, arising in the context of the plane Erdős distinct distance problem, to a second moment incidence estimate. This relies, in particular, on the generalisation of the Guth-Katz incidence bound by Solomon and Sharir.
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