There exists an edge coloring of $K_{10,10}$ with two colors such that there are at most $56$ monochromatic copies of $K_{3,3}$. Show that there exists an edge coloring of $K_{10,10}$ with two colors such that there are at most $56$ monochromatic copies of $K_{3,3}$.

There are $100$ edges in $K_{10,10}$ and $9$ edges in $K_{3,3}$.
If we consider $56$ copies of $K_{3,3}$, then there are a total of   $56*9 = 504$ edges which is much more than $100$. So there is certainly intersections between the monochromatic copies of $K_{3,3}$.

I am not getting any idea how to approach the problem.
Any hints will be appreciable. Thanks.

In fact, it is possible to have $0$ monochromatic copies of $K_{3,3}$.  For example, use colors $1$ and $2$ according to this $10 \times 10$ matrix, where each row corresponds to a left node and each column corresponds to a right node:
 2 1 1 1 1 2 2 2 1 2 
 1 1 1 2 2 2 2 2 1 1 
 1 1 1 2 1 2 2 1 2 2 
 1 2 1 1 2 2 1 2 2 1 
 1 1 2 1 2 1 2 1 2 2 
 1 2 2 2 1 2 1 2 1 2 
 2 2 2 1 2 1 1 2 1 2 
 2 2 1 2 1 1 2 1 1 2 
 2 1 2 2 1 1 1 1 2 1 
 2 2 1 1 2 1 1 1 1 1 

