Elementary versions of the Sylvester-Gallai theorem

Victor Pambuccian

Abstract: A Sylvester-Gallai (SG) configuration is a set $S$ of $n$ points such that the line through any two points of $S$ contains a third point in $S$. L. M. Kelly (1986) positively settled an open question of Serre (1966) asking whether an SG configuration in a complex projective space must be planar. N. Elkies, L. M. Pretorius, and K. J. Swanepoel (2006) have recently reproved this result using elementary means, and have proved that SG configurations in a quaternionic projective space must be contained in a three-dimensional flat. We point out that these results hold in a setting that is much more general than $\mathbb{C}$ or $\mathbb{H}$, and that, for each individual value of $n$, there must be truly elementary proofs of these results. Kelly's result must hold in projective spaces over arbitrary fields of characteristic 0 and the new result of Elkies, Pretorius and Swanepoel must hold in all quaternionic skew-fields over a formally real center.

Classification (MSC2000): 51A30, 03C35

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