Let $S=\{(i,j) \vert i,j=1,2,\ldots ,100\}$ be a set consisting of points on the coordinate plane. Each element of $S$ is colored one of four given colors. A subset $T$ of $S$ is called colorful if $T$ consists of exactly $4$ points with distinct colors, which are the vertices of a rectangle whose sides are parallel to the coordinate axes. Find the maximum possible number of colorful subsets $S$ can have, among all legitimate coloring patters.
What I thought: Now we prove that this is maximal. Let a rainbow L be a triple of cells $(O, X, Y)$ with $O$ and $X$ in the same row, $O$ and $Y$ in the same column, and the three colors all different. Observe that each rainbow rectangle is comprised of four rainbow Ls (and these Ls are different for different rectangles), so the number of rainbow rectangles is at most one-fourth the number of rainbow Ls.
One question that came to me at the beginning was: So, should $|T|=4$ or should it contain $4$ points with the desired property?
I hope for a solution! Soon, thank you!
[I think the answer is $9375000$ from some colleagues, but it may be wrong]
$2019$ CWMI