Note on disjoint faces in simple topological graphs

Abstract

We prove that every $n$-vertex complete simple topological graph generates at least $\Omega(n)$ 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 $k\geq 3$.