Two results about points, lines and planes

✍ Scribed by George Purdy

• Publisher: Elsevier Science
• Year: 1986
• Tongue: English
• Weight: 240 KB
• Volume: 60
• Category: Article
• ISSN: 0012-365X
• DOI: 10.1016/0012-365x(86)90014-2

✦ Synopsis

Given n points in three dimensional euclidean space, not all lying on a plane, let 1 be the number of lines determined by the points, and let p be the number of planes determined. We show that 1'3 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 2 0.