On Sets Defining Few Ordinary Planes

Let S be a set of n points in real three-dimensional space, no three collinear and not all co-planar. We prove that if the number of planes incident with exactly three points of S is less than K n 2 for some K = o ( n 1 / 7 ) then, for n sufficiently large, all but at most O ( K ) points of S are contained in the intersection of two quadrics. Furthermore, we prove that there is a constant c such that if the number of planes incident with exactly three points of S is less than 1 2 n 2 - c n then, for n sufficiently large, S is either a regular prism, a regular anti-prism, a regular prism with a point removed or a regular anti-prism with a point removed. As a corollary to the main result, we deduce the following theorem. Let S be a set of n points in the real plane. If the number of circles incident with exactly three points of S is less than K n 2 for some K = o ( n 1 / 7 ) then, for n sufficiently large, all but at most O ( K ) points of S are contained in a curve of degree at most four.