So I'm working on a problem that has to do with Ramsey Theory. We have $n$ guests at a christmas party. We know two things about them. In any group of three there are two people who do not know each other. In any group of $7$ there are two people who $\bf{do}$ know each other. At the end of the party, people give gifts to the ones that they know. Prove that the number of gifts given is at most $6n$.
I've been trying for a while to figure this problem out. I started out by looking at a group of $7$. We know that two people do know eachother. But can we say anything else about these people? I've tried breaking 6 of them into groups of 3, and saw that 4 gifts are given out amongst a group of 3. But I can't really get any where.
Hint: $R(3, 7) = 23$, so there are at most 22 guests. [I don't know how to prove this, but since you're saying it's related to Ramsey Theory, I suppose you can quote this fact.]
Hint: Mantel's Theorem as Vincent suggested.