For n≥3, n set of boys and girls has a stable matching (true or false) For some n ≥ 3 there exists a set of n boys, n girls, and preference lists for every boy
and girl such that every possible boy-girl matching is stable. If true, give a proof. If false, give a refutation.

This is obviously false as at n=3 I can find a unstable matching. But then I need to prove it for n≥3, no stable matching will appear. 

I tried doing it like this: Assume I keep adding a stable pair of couples starting at 3 pairs, no matter how many pairs I add the matching remains unstable as an unstable pair already exist in the group. But what if I added unstable couples, then I cannot promise at some n the matching might suddenly become stable. 

So is there a way to prove it?
You're asking about the Stable Marriage Problem.
Not only does there exist a stable setting, but you can also find it in ($n^2$) time!
First you fix a sex that "proposes", say men. Then every man, one after another, proposes to his first-choice woman. They accept provisionally. If, at a later point a woman, who has already accepted a proposal, gets an offer from a higher-preferred man, then she boots her first choice out. The rejected enters the loop again, then asking for his next-highest choice.
Now you only need to prove that the algorithm terminates, that everyone's married then and that the constellation is stable.
You're asking about the Stable Marriage Problem.
Not only does there exist a stable setting, but you can also find it in ($n^2$) time!
First you fix a sex that "proposes", say men. Then every man, one after another, proposes to his first-choice woman. They accept provisionally. If, at a later point a woman, who has already accepted a proposal, gets an offer from a higher-preferred man, then she boots her first choice out. The rejected enters the loop again, then asking for his next-highest choice.
Now you only need to prove that the algorithm terminates, that everyone's married then and that the constellation is stable.