Question: Let $X=[0,1]^d$ be the $d$-dimensional cube. Turn $X$ into a torus $T^d$ by identifying opposite facets. What is the minumum $(d-1)$-dimensional volume $f(d)$ of a subset $Y$ of $X$ which intersects every non-trivial cycle in $T^d$.
Answer: Taking $Y$ to be all points in the solid cube with one coordinate having value 1/2, gives you a set $Y$ that seperates all cycles and has $(d-1)$-dimensional volume equals $d$. It is not difficult to prove that $f(d) \ge C\sqrt d$. Guy Kindler, Ryan O’donnell, Anup Rao and Avi Wigderson proved the existence of $Y$ which seperates all cycles with $vol(Y) =O(\sqrt d)$. 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.