Some Sylvester-Gallai-Type Theorems for Higher-Dimensional Flats

Abstract

In response to an old question of Sylvester, Gallai proved that, for any finite set of points in the plane, not all contained in a single line, there is a line containing exactly two of them. This is known as the Sylvester-Gallai Theorem. In the language of matroid theory, which is used in this thesis, the Sylvester-Gallai Theorem says that every rank-3 real-representable matroid has a two-point line. Independently of Gallai and around the same time, Melchior proved the stronger result that, in a rank-3 real-representable matroid, the average number of points in a line is less than three. The purpose of this thesis is to prove theorems of this type for flats of higher dimension.

The Sylvester-Gallai Theorem was generalized to higher dimensions by Hansen, who proved that, for any positive integer k, a simple real-representable matroid of rank at least 2k-1 has a rank-k independent flat. We will prove a theorem of this type for matroids representable over any given finite field by characterizing the "unavoidable flats" for matroids in this class. In particular, we show that any simple matroid of sufficiently high rank representable over a given finite field has a rank-k flat which is either independent or a projective or affine geometry. As a corollary, we will get the following Ramsey theorem: for any two-colouring of the points of a matroid of sufficiently high rank representable over a given finite field, there is a monochromatic rank-k flat.

The rest of the thesis concerns generalizing Melchior’s theorem to higher dimensions. For planes, we prove that, in a real-representable matroid M of rank at least four, the average plane-size is at most an absolute constant, unless M is a direct sum of lines (in which case the average plane-size could be arbitrarily large). Generalizing this result to hyperplanes of arbitrary rank k, we prove that, if M is a simple rank-(k+1) real-representable matroid whose average hyperplane-size is greater than some constant depending only on k, then all but a constant number b_k of the points of M are covered by a small collection of flats of relatively low rank. We also show that this constant b_k is best-possible.

Melchior’s Theorem also implies that the average size of a flat (of any rank) in a rank-3 real-representable matroid is at most an absolute constant. We prove that, for any positive integer r, the average size of a flat in a rank-r real-representable matroid is at most some constant depending only on r.