Question 13
Given positive integers $n$ and $k$, $n \gt k^2 \gt 4$. In an $n \times n$ grid, any $k$ squares in distinct rows and distinct columns are called a $k$-set. Find the largest positive integer $N$ satisfying that: one can choose $N$ squares of the $n \times n$ grid and colour them in a certain way, such that for any coloured $k$-set, some two squares have the same colour, and some two squares have different colours.