A note on the largest bipartite subgraph in point-hyperplane incidence graphs

TL;DR

This paper investigates the size of the largest complete bipartite subgraph in point-hyperplane incidence graphs in higher dimensions, providing bounds that close previous gaps in four and five dimensions.

Contribution

It extends bounds on the largest bipartite subgraph in point-hyperplane incidence graphs to four and five dimensions, improving upon prior results limited to three dimensions.

Findings

Established bounds for the largest bipartite subgraph in 4D and 5D point-hyperplane incidences

Closed the gap in incidence bounds in four and five dimensions

Results are tight up to logarithmic factors

Abstract

Given $m$ points and $n$ hyperplanes in $\mathbb{R}^d$, if there are many incidences, we expect to find a big cluster $K_{r,s}$ in their incidence graph. Apfelbaum and Sharir found lower and upper bounds for the largest size of $rs$, which only match in three dimensions. In this paper we close the gap in four and five dimensions, up to some logarithmic factors.