The lines and planes connecting the points of a finite set

Abstract

0.1. Not later than 1933 I made the following conjecture, originally in the form of a statement on the minors of a matrix(1). T<¡. Any n points in d-space that are not on one hyper plane determine at least n connecting hyperplanes.Ti is trivial. It is easy to see (4.3) that Td is a consequence of Td_i and Ud-Any n points in d-space that are not on one hyper plane determine at least one ordinary hyperplane, that is, a connecting hyperplane on which all but one of the given points are on one linear (d -2)-space.In particular, T2: n not collinear points are connected by at least n straight lines, is true if U2 holds: for n not collinear points there is a straight line through only two of them. Now the nine inflexions of a plane cubic show that U2 does not hold in the complex plane. Nevertheless H. Hanani gave in 1938 a combinatorial proof of T2 for every (not only the real) projective plane. A greatly simplified version of this proof is given in 4.4. In 1939 A. Robinson proved U2 for the real plane; in 1943 I found another very short proof (respectively the second and first proof in 1

A further theorem of the Sylvester type

“…The present paper obtains the stronger conclusion of Theorem II from the weaker hypotheses of Theorem I. For a more elaborate bibliography and historical comment on problems of the Sylvester type see [2], [3], and [4].…”

Combinatorial geometry and actions of compact Lie groups

“…In combinatorial geometry the theorem of Sylvester-Gallai has passed through stages of reproving [10] and generalization, notably by Motzkin [14] and Sten Hansen [11] …”

Beitrag zur kombinatorischen Inzidenzgeometrie

“…Es sollen also weiter keine (n 1) Punkte auf derselben Gerade oder demselben Kreis liegen. MOTZKIN [5] hat gezeigt: Wenn keine (n-1) der gegebenen n Punkte in der Ebene auf derselben Gerade liegen, dann k6nnen nicht alle Geraden 2. Ordnung mit demselben Punkt inzidieren, die dutch diese n Punkte definiert sind.…”