Combinatorics and Optimization Professor talk - Bruce Richter

Sylvester's 4-point problem and straight line drawings of the complete graph K_n

Abstract: In the 1880's, Sylvester raised the following question: if we
pick 4 points at random from the Euclidean plane, what is the probability
that they make a convex quadrilateral? Equivalently, what is the
probability that one point is inside the triangle formed by the other
three? We will make this question more precise in the talk.

A problem in graph theory is to determine, over all sets of n points in the
plane, no three collinear, the smallest number f(n) of crossings of the n
choose 2 straight line segments joining all pairs of these n points. It is
easy to see that no crossings are required for n <= 4, f(5)=1, f(6)=3,
f(7)=9.