On the Number of Incidences When Avoiding an Induced Biclique in Geometric Settings

TL;DR

This paper establishes new bounds on the number of incidences between points and geometric regions, specifically axis-parallel boxes, under certain restrictions, improving previous results and matching known lower bounds.

Contribution

It provides a new upper bound on incidences avoiding a biclique in geometric settings, and extends the analysis to various shapes like halfspaces, pseudodisks, and fat triangles.

Findings

Bound of $O(k n(rac{ ext{log} n}{ ext{log} ext{log} n})^{d-1})$ for points and boxes in $ extbf{R}^d$

Linear bounds for incidences with halfspaces in 2D and 3D

Near linear bounds for shapes with low union complexity

Abstract

Given a set of points $P$ and a set of regions $\mathcal{O}$, an incidence is a pair $(p,o ) \in P \times \mathcal{O}$ such that $p \in o$. We obtain a number of new results on a classical question in combinatorial geometry: What is the number of incidences (under certain restrictive conditions)? We prove a bound of $O\bigl( k n(\log n/\log\log n)^{d-1} \bigr)$ on the number of incidences between $n$ points and $n$ axis-parallel boxes in $\mathbb{R}^d$, if no $k$ boxes contain $k$ common points, that is, if the incidence graph between the points and the boxes does not contain $K_{k,k}$ as a subgraph. This new bound improves over previous work, by Basit, Chernikov, Starchenko, Tao, and Tran (2021), by more than a factor of $\log^d n$ for $d >2$. Furthermore, it matches a lower bound implied by the work of Chazelle (1990), for $k=2$, thus settling the question for points and boxes. We also study several other variants of the problem. For halfspaces, using shallow cuttings, we get a linear bound in two and three dimensions. We also present linear (or near linear) bounds for shapes with low union complexity, such as pseudodisks and fat triangles.