Show that these 100 squares are the same color. The squares of an infinite square graph paper are colored in 3 colors. Show that there exist 100 rows and 100 columns such that all squares in the intersections of the said columns (100 square) have the same color.
I am stuck the question is so ambiguous and that there are so many cases. Where do I start?
 A: Suppose the colors are blue, green, and red.
Choose any row and a color that appears an infinite number of times in that row. Reduce the grid to only those columns that intersect the row with that color. Now we have an infinite grid with a monochromatic row, say the color is blue.
If blue now appears an infinite number of times in another row, repeat the process to get two blue rows. Similarly, if there then is a third row where blue appears infinitely many times, and so on. If we eventually find 100 blue rows then we can choose our monochromatic 100 by 100 blue box.
If not, then blue only appears finitely many times in every remaining row. There must be a color that appears infinitely many times in these rows, so choose a row and suppose green appears infinitely many times. Reduce to get a green row, etc. If we get to 100 green rows, we get a monochromatic 100 by 100 green box.
If not, then all remaining rows only contain a finite number of blue and green squares. Therefore, we can find a monochromatic 100 by 100 red box.
