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

Given a set of points \(P\) and a set of regions \(\mathcal {O}\), an incidence is a pair \((p,\mathcalligra {o}) \in P\times \mathcal {O}\) such that \(p\in \mathcalligra {o}\). 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 et al. (Forum Math Sigma 9:59, 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 (J ACM 37(2):200–212, 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.