# Combinatorics Seminar

When: Sunday, April 30, 10am

Where: Schreiber 309

Speaker: Rom Pinchasi (Technion)

Title: On the Sylvester Graph of points in the plane

## Abstract:

Let $P$ be a finite set of points in the plane. An ordinary line determined by $P$ is a line passing through precisely two points of $P$. It is a classical result (Gallai-Sylvester theorem) that unless $P$ is collinear it must determine ordinary lines. We call a point $x \in P$ ordinary if there is an ordinary line through $x$. The Sylvester Graph of $P$ is the graph whose vertices are the ordinary points of $P$ and we connect two points by an edge if they determine an ordinary line. We show that if the Sylvester Graph of $P$ is a complete graph, then either $P$ is in general position, or it is the so called 'failed Fano configuration' (a special configuration of $7$ points). This proves a 120 years old conjecture of Sylvester.

This is a joint work with Eyal Ackerman.

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