Two results about points, lines and planes

Abstract

Given n points in three dimensional euclidean space, not all lying on aplane, let l be the number of lines determined by the points, and let p be the number of planes determined. We show that l 2⩾ cnp, where c>0. This is the weak version of the so-called Points-Lines-Planes conjecture (a conjecture of considerable interest to combinatorialists) being an instance of the conjectured log-concavity of the Whitney numbers. We also show that there is always a point incident with at least cl planes, where c>0, provided that the n points do not all lie on two skew lines. This result lends support to our conjecture, published in 1981, that n − 1 + p + 2 ⩾ 0.