The Erdos-Szekeres theorem guarantees empty convex polygons in sufficiently large point sets. Gerken proved empty convex hexagons exist. These results live in Euclidean geometry — points in the plane, convexity, area.
Aichholzer, Garcia, Parada, Scheucher, and Vogtenhuber extend the theory to simple drawings of complete graphs. A simple drawing places vertices in the plane with edges as curves crossing at most once. The “polygons” become plane k-cycles — cycles whose edges don't cross each other.
They prove empty 4-cycles exist in all simple drawings of K_n for large n. An empty 4-cycle is a plane quadrilateral containing no other vertex in its interior (defined via the Jordan curve theorem). For convex drawings, they generalize Gerken's hexagon theorem: empty plane hexagons always exist.
The extension reveals that the classical empty polygon theorems were topological all along. The proofs adapt Ramsey-type arguments from metric geometry to a setting where “inside” is defined by topology (Jordan curve) rather than geometry (convex hull). The convexity that seemed essential to the Euclidean results was structural scaffolding — the real content was combinatorial.
General simple drawings may lack empty hexagons — the convex structure provides rigidity the general case doesn't have. The boundary between what topology guarantees and what geometry adds is precisely the gap between 4-cycles (topological) and 6-cycles (geometric).