Random half-integral polytopes

We show that polytopes obtained as the convex hull of a random set of half-integral points of the 0/1 cube have rank as high as Ω ( log n / log log n ) with positive probability—even if the size of the set relative to the total number of half-integral points of the cube tends to 0. The high rank is due to certain obstructions. We determine the exact threshold number, when those cease to exist.