On a generalization of the Gallai-Sylvester theorem

Abstract

It is shown that if S ⊂ ℝd, aff S=aff ℝd, and every hyperplane spanned by (a subset of)S misses fewer than k points of S(k≥2), then (a) #S≤km if d=2 m-1 is odd and (b) #S≤km+1 if d=2 m is even. We also fully describe the extreme sets for which equality holds in (a) or in (b). For odd d the proofs are later modified to purely algebraic ones, and carry over to {Mathematical expression}, where {Mathematical expression} is an arbitrary field. For even d, (b) is generally not true when {Mathematical expression}, but we prove some weaker inequalities that do hold over arbitrary fields. © 1992 Springer-Verlag New York Inc.