Title: Short edges in topological graphs

Speaker: Andrew Suk, UCSD

Date and time: 6pm (New York time), Tuesday, October 17, 2023

Place: WWH1314 (room 1314 Warren Weaver Hall, 251 Mercer Street)

...and on Zoom, details on the seminar mailing list

Let $h(n)$ be the minimum integer such that every complete $n$-vertex simple topological graph contains an edge that crosses at most $h(n)$ other edges. In 2009, KynĨl and Valtr showed that $h(n)=O(n^2/\log^{1/4}n)$, and in the other direction, gave constructions showing that $h(n)=\Omega(n^{3/2})$. In this talk, I will sketch a proof showing that $h(n)=O(n^{7/4})$. The proof is based on a variant of Chazelle-Welzl's matching theorem for set systems with bounded VC-dimension, which may be of independent interest. I will also discuss several other related problems.