Three Counterexamples on Semigraphoids
  Semigraphoids are combinatorial structures that arise in statistical learning theory. They are equivalent to convex rank tests and to polyhedral fans that coarsen the reflection arrangement of the symmetric group. We resolve two problems on semigraphoids posed in Studeny's book, and we answer a related question by Postnikov, Reiner, and Williams on generalized permutohedra. We also study the semigroup and the toric ideal associated with semigraphoids. 
Raymond Hemmecke, Jason Morton, Anne Shiu, Bernd Sturmfels, and Oliver   Wienand