Question 20
We call two people a pair of friends if they know each other, and we call them a non-friend pair if they do not know each other (being friends or not being friends is always mutual). Let $m, n$ be natural numbers. Find the smallest natural number $k$ such that the following holds: in every group of $k$ people there are always $2 m$ people who form $m$ disjoint pairs of friends, or there are $2 n$ people who form $n$ disjoint non-friend pairs.