On Geometric Bipartite Graphs with Asymptotically Smallest Zarankiewicz Numbers

TL;DR

This paper investigates the Zarankiewicz problem in low-dimensional geometric bipartite graphs, revealing bounds that distinguish between Ferrers dimensions three and four, and providing tight bounds for specific graph classes like chordal bigraphs and grid intersection graphs.

Contribution

It establishes new bounds for Zarankiewicz numbers in geometric bipartite graphs, highlighting differences based on Ferrers dimension and offering tight bounds for certain graph classes.

Findings

Bound of $9n(k-1)$ for Ferrers dimension three graphs

Lower bound of $rac{n k rac{ ext{log} n}{ ext{log} ext{log} n}}$ for Ferrers dimension four graphs

Tight upper bounds for chordal bigraphs and grid intersection graphs

Abstract

This paper considers the \textit{Zarankiewicz problem} in graphs with low-dimensional geometric representation (i.e., low Ferrers dimension). Our first result reveals a separation between bipartite graphs of Ferrers dimension three and four: while $Z(n;k) \leq 9n(k-1)$ for graphs of Ferrers dimension three, $Z(n;k) \in \Omega\left(n k \cdot \frac{\log n}{\log \log n}\right)$ for Ferrers dimension four graphs (Chan & Har-Peled, 2023) (Chazelle, 1990). To complement this, we derive a tight upper bound of $2n(k-1)$ for chordal bigraphs and $54n(k-1)$ for grid intersection graphs (GIG), a prominent graph class residing in four Ferrers dimensions and capturing planar bipartite graphs as well as bipartite intersection graphs of rectangles. Previously, the best-known bound for GIG was $Z(n;k) \in O(2^{O(k)} n)$, implied by the results of Fox & Pach (2006) and Mustafa & Pach (2016). Our results advance and offer new insights into the interplay between Ferrers dimensions and extremal combinatorics.