The Sylvester-Gallai theorem, colourings and algebra

The Sylvester-Gallai Theorem, the Monochrome Line Theorem and Generalizations Report for a Seminar on the Sylvester-Gallai Theorem

A q-Analogue of the Centralizer Construction and Skew Representations of the Quantum Affine Algebra

We prove an analogue of the Sylvester theorem for the generator matrices of the quantum affine algebra Uq(ĝln). We then use it to give an explicit realization of the skew representations of the quantum affine algebra. This allows one to identify them in a simple way by calculating their highest weight, Drinfeld polynomials and the Gelfand–Tsetlin character (or q-character). We also apply the qu...

The Sylvester-Chvatal Theorem

The Sylvester-Gallai theorem asserts that every finite set S of points in two-dimensional Euclidean space includes two points, a and b, such that either there is no other point in S is on the line ab, or the line ab contains all the points in S. V. Chvátal extended the notion of lines to arbitrary metric spaces and made a conjecture that generalizes the Sylvester-Gallai theorem. In the present ...

Improved rank bounds for design matrices and a new proof of Kelly's theorem

We study the rank of complex sparse matrices in which the supports of different columns have small intersections. The rank of these matrices, called design matrices, was the focus of a recent work by Barak et. al. [BDWY11] in which they were used to answer questions regarding point configurations. In this work we derive near-optimal rank bounds for these matrices and use them to obtain asymptot...

A Reverse Analysis of the Sylvester-Gallai Theorem

Reverse analyses of three proofs of the Sylvester-Gallai theorem lead to three different and incompatible axiom systems. In particular, we show that proofs respecting the purity of the method, using only notions considered to be part of the statement of the theorem to be proved, are not always the simplest, as they may require axioms which proofs using extraneous predicates do not rely upon.