Graph theory question, probably connected to Ramsey Theorem Is the following statement true or false? For every $n > 0$, such $N$ exists, that no matter how we "color" all of the subsets of the set with $N$ elements(we only use two colors), No matter how we color the subsets of the N-elemented set with two colors, we can always find an n-elemented subset, whose subsets have the same colour, sorry I wrote it wrong. I am just confused how should I start this task, my guess is false, but I can't really begin anything. Any help appreciated :)

The statement is false for all $n > 0$ because one can color the subsets according to the parity of their sizes, that is, if the size of the subset is odd, then color it blue, otherwise red.