Given $20$ people. Show that there exists a group of $4$ which either everyone in the group know each other, or no one know each other at all.
Ok, so I've read and found that $R(4,4) = 18$ on Wikipedia
And I found that they've prove by bounding the $R(4,4)$ value using recurrence relation $R(r, s) ≤ R(r − 1, s) + R(r, s − 1) − 1$ , which yields $R(4,4) \leq 18$. Then, they construct an counter-example (here) and show that $R(4,4) \gt 17$. Hence, $R(4,4) = 18$.
Now, I've wonder , if we weaken the value to $20$ , can we prove the question above by using some elementary knowledge, such as pigeonhole principal?
Thanks in advance!
PS : If this question was asked already, please show me the link. Thank you!