Prove that among $ 9 $ people, there are $ 3 $ who know each other or $ 4 $ who do not know each other.
I began by trying to prove $ R_9 \rightarrow R_3, R_4 $. Let $ S $ be a set of $ 9 $ points in the plane, no three of which are collinear. Let the lines joining these points be coloured either red or blue (red denotes a relation, blue denotes none). Consider any one of the points and the $8$ line segments joining this point (say $ P $) with the other points on the plane. By the pigeon-hole principle, at least $4$ of them are either red or blue. WLOG assume four red lines meet the other points at $A$, $B$, $C$ and $D$ respectively. Consider the lines joining these points. If all of them are blue, then we have a blue $K_4$. If one of them is red, then we have a red $K_3$.
Is my reasoning correct?
I'm quite new to Ramsey's theorem and I'm still a high school sophomore so please answer accordingly so I can comprehend.