The Sylvester-Chvatal Theorem

Polygon Partitions

In 1973, Victor Klee posed the problem of determining the minimum number of guards sufficient to cover the interior of an n-wall art gallery room (Honsberger 1976). He posed this question extemporaneously in response to a request from Vasek Chvatal (at a conference at Stanford in August) for an interesting geometric problem, and Chvatal soon established what has become known as "Chvatal's Art G...

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

A Note on Perfectly orderable Graphs

A natural way to colour the vertices of a graph is: (i) to impose a linear order < on the vertices, and (ii) to scan the vertices in this order, assigning to each vertex c(,j) the smallest positive integer assigned to no neighbour v(k) of o(j) with z>(k) < t:(,j). This heuristic algorithm is called the greedy colouring algorithm, or the sequential colouring algorithm. One may ask the following ...

Louis J . Gross , Scott M . Duke - Sylvester , and Mark Palmer

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.