Discrete Comput Geom (2018) 60:220–253
On Sets Deﬁning Few Ordinary Planes
Received: 30 August 2016 / Revised: 7 August 2017 / Accepted: 4 September 2017 /
Published online: 19 September 2017
© Springer Science+Business Media, LLC 2017
Abstract 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 Kn
for some K = o(n
) 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
− cn 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 Kn
some K = o(n
) then, for n sufﬁciently large, all but at most O(K ) points of S are
contained in a curve of degree at most four.
Keywords Ordinary planes · Eight associated points theorem · Sylvester–Gallai ·
Mathematics Subject Classiﬁcation 51M04 · 52C35
Editor in Charge: János Pach
The author acknowledges the support of the Project MTM2014-54745-P of the Spanish Ministerio de
Economía y Competitividad.
Departament de Matemàtiques, Universitat Politècnica de Catalunya, Jordi Girona 1-3 Mòdul C3,
Campus Nord, 08034 Barcelona, Spain