Variations on the Gallai-Milgram theorem

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

Hypergraph Extensions of the Erdos-Gallai Theorem

Our goal is to extend the following result of Erd˝ os and Gallai for hypergraphs: Theorem 1 (Erd˝ os-Gallai [1]) Let G be a graph on n vertices containing no path of length k. Then e(G) ≤ 1 2 (k − 1)n. Equality holds iff G is the disjoint union of complete graphs on k vertices. We consider several generalizations of this theorem for hypergraphs. This is due to the fact that there are several po...

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...

An Erd os-Gallai Theorem For Matroids

Erd os and Gallai showed that for any simple graph with n vertices and circumference c it holds that |E(G)| ≤ 2 (n−1)c.We extend this theorem to simple binary matroids having no F7-minor by showing that for such a matroid M with circumference c(M)≥ 3 it holds that |E(M)| ≤ 1 2 r(M)c(M). Mathematics Subject Classi cation (2000). Primary 05D15; Secondary 05B35.