Monochromatic Rectangle of a 2-Colored 8 by 8 Lattice Grid

Question

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 $7 \times 7$?

Answer 1

The $9\times3$ proof you linked to actually works just as well for a $7\times3$ grid. The point is, each row of three vertices contains a monochromatic pair: $2$ possibilities for the color, $3$ possibilities for the location. (If all $3$ vertices are the same color, choose one pair arbitrarily.) There are $2\times3=6$ patterns, so one of them must repeat.