Unbalanced Zarankiewicz problem for bipartite subdivisions with applications to incidence geometry

TL;DR

This paper investigates the unbalanced Zarankiewicz problem for bipartite subdivisions, establishing bounds on the linear threshold for certain graphs and applying these results to incidence geometry problems involving points and lines in the complex plane.

Contribution

It determines the linear threshold for the complete bipartite subdivision graph and applies this to derive new bounds in incidence geometry and distance problems.

Findings

Linear threshold of $K_{s,t}'$ is at most $2 - 1/s$.

Graphs with parts unbalanced by a factor less than the threshold contain many edges.

Applications include bounds on incidences and distances in the plane.

Abstract

For a bipartite graph $H$, its linear threshold is the smallest real number $\sigma$ such that every bipartite graph $G = (U \sqcup V, E)$ with unbalanced parts $|V| \gtrsim |U|^\sigma$ and without a copy of $H$ must have a linear number of edges $|E| \lesssim |V|$. We prove that the linear threshold of the complete bipartite subdivision graph $K_{s,t}'$ is at most $\sigma_s = 2 - 1/s$. Moreover, we show that any $\sigma < \sigma_s$ is less than the linear threshold of $K_{s,t}'$ for sufficiently large $t$ (depending on $s$ and $\sigma$). Some geometric applications of this result are given: we show that any $n$ points and $n$ lines in the complex plane without an $s$-by-$s$ grid determine $O(n^{4/3 - c})$ incidences for some constant $c > 0$ depending on $s$; and for certain pairs $(p,q)$, we establish nontrivial lower bounds on the number of distinct distances determined by $n$ points in the plane under the condition that every $p$ points determine at least $q$ distinct distances.