Kelly (1986) answered Serre's question affirmatively; Elkies, Pretorius & Swanepoel (2006) simplified Kelly's proof, and proved analogously that in spaces with quaternion coordinates, all Sylvester–Gallai configurations must lie within a three-dimensional subspace.
Motzkin observed that, for these parameters to define a Sylvester–Gallai design, it is necessary that b > 2, that p < ℓ (for any set of non-collinear points in a projective space determines at least as many lines as points) and that they also obey the additional equation For, the left hand side of the equation is the number of pairs of points, and the right hand side is the number of pairs that are covered by lines of the configuration.
Sylvester–Gallai designs that are also projective configurations are the same thing as Steiner systems with parameters ST(2,b,p).
Motzkin listed several examples of small configurations of this type: Boros, Füredi & Kelly (1989) and Bokowski & Richter-Gebert (1992) studied alternative geometric representations of Sylvester–Gallai designs, in which the points of the design are represented by skew lines in four-dimensional space and each line of the design is represented by a hyperplane.
Kelly & Nwankpa (1973) more generally classified all non-collinear Sylvester–Gallai configurations and Sylvester–Gallai designs over at most 14 points.