Question: Let be the -dimensional cube. Turn into a torus by identifying opposite facets. What is the minumum -dimensional volume of a subset of which intersects every non-trivial cycle in .
Answer: Taking to be all points in the solid cube with one coordinate having value 1/2, gives you a set that seperates all cycles and has -dimensional volume equals . It is not difficult to prove that . Guy Kindler, Ryan O’donnell, Anup Rao and Avi Wigderson proved the existence of which seperates all cycles with . A simpler argument was found by Noga Alon and Boaz Klartag. For an even simpler treatement of this result along with several discrete analogs see this paper by Noga.