Prove that if there are 100 people who each know 67 others among this 100, there are 4 people who all know each other.

Question

Knowing is mutual (if A knows B, then B knows A).

I'm stuck - I don't know where to begin. Judging by the format of the problem, I'm thinking pigeonhole probably.

Answer 1

We want to prove the graph has a $K_4$. The graph has more edges than $K_{33,33,34}$ so by Turan's theorem it is immediate.

Alternative solution:

Pick two adjacent vertices $v_1$ and $v_2$, clearly they have at least $34$ common neighbours. Pick a vertex $v_3$ that is a common neighbour, clearly one of its neighbours must also be among the $34$ common neighbours of $v_1$ and $v_2$ if we call this vertex $v_4$ then $v_1,v_2,v_3,v_4$ form a $K_4$.