Sylvester-Gallai type theorems for approximate collinearity

Sylvester-Gallai Theorems for Complex Numbers and Quaternions

A Sylvester-Gallai (SG) configuration is a finite set S of points such that the line through any two points in S contains a third point of S. According to the Sylvester-Gallai Theorem, an SG configuration in real projective space must be collinear. A problem of Serre (1966) asks whether an SG configuration in a complex projective space must be coplanar. This was proved by Kelly (1986) using a d...

Fractional Sylvester-Gallai Theorems∗

We prove fractional analogs of the classical Sylvester-Gallai theorem. Our theorems translate local information about collinear triples in a set of points into global bounds on the dimension of the set. Specifically, we show that if for every points v in a finite set V ⊂ C, there are at least δ|V | other points u ∈ V for which the line through v, u contains a third point in V , then V resides i...

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

Identity Testing for Depth-3 Circuits with Small Fan-in

4 History of the Sylvester-Gallai Problem 6 4.1 Melchior’s Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 4.2 Motzkin’s Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 4.3 Kelly and Moser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 4.4 Green and Tao . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 4.5 How to describe 3-color ...

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 ...