Suppose you have a subset $S$ of the vertices of a planar graph which contains at least one vertex from every face. Then $S$ must have at least half of the vertices, and for some planar graphs every such $S$ must have at least half of the vertices. We believe this extends to higher dimensions, but don’t really know why, and have found some situational evidence (but also some counter-evidence). This is based on joint work with Michael Dobbins and Seunghun Lee.