On Geometric Bipartite Graphs with Asymptotically Smallest Zarankiewicz Numbers

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.