This is actually not a homework problem, but I'd like it to be treated as if it were one. I am not looking for a solution, I would just like to have some hints on how to start.
I am trying to solve the third problem here. I managed to solve the first two, so I think I understand what a probabilistic argument is. Let me rewrite the question here. I would like to prove that for any given set $S \subset \mathbb{Z}\setminus\{0\}$ of size $n$, there must be a pair of disjoint subsets $A$ and $B$, such that $|A| + |B| > 2 n / 3$ and both $A$ and $B$ are sum-free (each does not contain three elements where one is the sum of the other two).
I am really stuck in figuring out the "right" random variable to look at. Any (non-spoiler) hints are very much appreciated!