Moment curves have been used for several applications in discrete geometry including cyclic polytopes, the no-three-in-line problem, and a geometric proof of the chromatic number of Kneser graphs.
[4] In the Euclidean plane, it is possible to divide any area or measure into four equal subsets, using the ham sandwich theorem.
However, this result does not generalize to five or more dimensions, as the moment curve provides examples of sets that cannot be partitioned into 2d subsets by d hyperplanes.
This lemma, in turn, can be used to calculate the chromatic number of the Kneser graphs, a problem first solved in a different way by László Lovász.
The main idea is to choose a prime number p larger than n and to place vertex i of the graph at coordinates Then a plane can only cross the curve at three positions.
A similar construction using the moment curve modulo a prime number, but in two dimensions rather than three, provides a linear bound for the no-three-in-line problem.