On a $7 \times 7$ grid of points $(1,1), (1,2), \dots, (7,7)$, show that any coloring of the vertices with two different colors will result in at least one set of four points that form the vertices of a rectangle and are all the same color.
With an infinitely large grid we can take 9 vertical lines intersected with 3 horizontal ones and use Pigeonhole (see here for more info). Can we adapt this proof somehow for an $7 \times 7$?