On the Number of Incidences when Avoiding the Klan

Abstract

We prove a bound of O(k(n + m) log d−1 ) on the number of incidences between n points and m axis parallel boxes in R d , if no k boxes contain k common points. That is, 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 a factor of log d n, for d > 2.We also study the variant of the problem for points and halfspaces, where we use shallow cuttings to get a near linear bound in two and three dimensions.