Is there a "Bipartite" Szemeredi-Trotter theorem?

Question

One version of the Szemeredi-Trotter theorem states the following:

Given a set of $L$ lines in the plane, the number of points incident to at least $k$ lines is bounded above by a constant times $L/k + L^2/k^3$.

This version of the theorem can be found here in notes written by Adam Hesterberg for Larry Guth's polynomial method class: http://math.mit.edu/~lguth/PolyMethod/lect6.pdf.

Let's suppose instead that I have a collection of $R$ red lines and $B$ blue lines, where $R$ is much smaller than $B$ (like $R$ is about $B^{1/2}$ or so). Assume that none of the red lines are parallel. I would like a bound for the number of points lying on at least one red line that are incident to at least $k$ blue lines. It's possible to use the Szemeredi-Trotter theorem to get a bound of $B/k + B^2/k^3$ by just ignoring the condition that the points need to lie along a red line. Does anyone know if a better bound is possible?

Answer 1

I will describe a construction in the dual setting.

Take a $(k+1)\times n/k$ size grid of points. Color the top row red, the rest blue, so we have $n$ red points and $k$ blue points. The number of lines that contain one point of each row is $\Theta(n^2/k^3)$.

So translating it back to your problem, this gives a construction of $B^2/k^3$ if $R=k$, so this shows that your bound is sharp if $R\ge k$.