Note on disjoint faces in simple topological graphs

Abstract

We prove that every n-vertex complete simple topological graph generates at least $\Omega \left(n\right)$ pairwise disjoint 4-faces. This improves upon a recent result by Hubard and Suk. As an immediate corollary, every n-vertex complete simple topological graph drawn in the unit square generates a 4-face with area at most $O(1/n)$. This can be seen as a topological variant of the Heilbronn problem for quadrilaterals. We construct examples showing that our result is asymptotically tight. We also discuss the similar problem for k-faces with arbitrary $k\ge 3$.