2023/08/09 by Zeng, Ji
#Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2308.04742
We prove that every n-vertex complete simple topological graph generates at least Ω(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≥ 3.