Combinatorics Seminar

Consider a finite set of points in the real plane. The Sylvester-Gallai theorem states that if the line through every pair of points contains a third point in the set then all points lie on the same line. In this work we prove a version of this theorem in which points are replaced with k-dimensional subspace and three subspaces are `collinear' if they are contained in the span of a single 2k-dimensional space. Under the (necessary) assumption that every pair of subspaces has trivial intersection, we show that: if every pair of spaces is in a collinear triple then all spaces are contained in dimension at most O(k^4). We also prove average versions of the theorem when only many pairs are in collinear triples. The main tool in the proof is a high dimensional generalization of a result of Barthe showing that one can find a change of basis that makes the average angle between spaces large on average. (Joint work with Guangda Hu).