  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.

Computational Geometry