The Sylvester-Chvatal Theorem

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 ...

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

The Sylvester-Gallai theorem, colourings and algebra

Our point of departure is the following simple common generalisation of the Sylvester-Gallai theorem and the Motzkin-Rabin theorem: Let S be a finite set of points in the plane, with each point coloured red or blue or with both colours. Suppose that for any two distinct points A, B ∈ S sharing a colour there is a third point C ∈ S, of the other colour, collinear with A and B. Then all the point...

Elementary Versions of the Sylvester-Gallai Theorem

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 m...

Sylvester-Gallai Theorem and Metric Betweenness

Sylvester conjectured in 1893 and Gallai proved some forty years later that every finite set S of points in the plane includes two points such that the line passing through them includes either no other point of S or all other points of S. There are several ways of extending the notion of lines from Euclidean spaces to arbitrary metric spaces. We present one of them and conjecture that, with li...