This is a dublicate from stackexchange:


Consider two families of hyperplanes $F\_1$ and $F\_2$ in $\mathbb{R}^d$ both containing $n$ hyperplanes. We have that for all $f \in F\_1$ and $g \in F\_2$ that $f$ and $g$ intersect. Further we know that for each point $p \in \mathbb{R}^d$ we have that at most $c \cdot n$ hyperplanes from each family contain this point, $ c < 1$.


The second family induces $d-2$ dimensional planes on the hyperplanes of the first family. I want to show that there are at least $\epsilon \cdot n$ hyperplanes in the first family on which at least $\tau \cdot n$ planes are induced.


What I am able to show is: If there is a plane in which $k \cdot n$ hyperplanes form the second family intersect then, there this plane ist contained in at most $c \cdot n$ hyperplanes from the first family. Consider the $(1-c)n$ hyperplanes of the first family which do not contain the plane, than on each of them we get $k \cdot n$ induced planes.


On the other hand it is clear if at most a constant number, $k$, of hyperplanes intersect in one plane we get $\frac{n}{k}$ induced planes on each of the hyperplanes in the first family.


So somehow I can show the extremal cases but I'm stuck with the ones in between. Any help or suggestions?


**Edit:**
Let $n\_H$ denote the number of $d-2$ dimensional planes induced on the plane $H \in F\_1$. Further let $f\_{H,1},...,f\_{H,n\_H}$ denote this induced planes and $S(f\_{H,i}):=\{K \in F\_2 \mid f\_{H,i} \subset K\}$. Then for $H, H' \in F\_1$ we have:


\begin{equation}
\frac{n}{d} = \sum\_{i=1}^{n\_H} \sum\_{j=1}^{n\_K} \vert S(f\_{H,i}) \cap S(f\_{H',j})\vert \leq c \cdot \frac{n}{d}+ n\_{H} \cdot n\_{H'}
\end{equation}


Since $\vert S(f\_{H,i}) \cap S(f\_{H',j})\vert \leq 1$ for all $f\_{H\_i} \neq H \cap H'$ and $\vert S(f\_{H,i}) \cap S(f\_{H',j})\vert \leq c \cdot \frac{n}{d}$ if $f\_{H,i} = f\_{H',j}=H \cap H'$. Therefore $(1-c)\frac{n}{d} \leq n\_{H'} \cdot n\_H$. This holds for all $H, H' \in F\_1$ so $n\_H \gg \sqrt{n}$.



Counting intersections of hyperplanes
As by my comment above, it is enough to solve the problem for $d=2$.  

Using point-line duality, we get the following:  

Given $n$ red and $n$ blue points in the plane, such that no line contains more than $cn$ points, prove that through at least $\varepsilon n$ of the red points there are at least $\tau n$ lines that each go through some blue point.


This can be proved similarly to Beck's theorem that can be [found here](https://en.wikipedia.org/wiki/Beck%27s_theorem_(geometry)#Proof_2).  

More precisely, there are $n^2$ red-blue point pairs.  

At most $O(cn^2)$ of the lines of these pairs contains at least $1/c$ points (see the linked proof).  

As there are no lines with $cn$ points, most pairs will contain at most $1/c$ points, which implies that there are at least $\varepsilon n$ red points with $\tau n$ lines through each with some blue point.


