I try to understand the proof of [Ramsey's Theorem](https://proofwiki.org/wiki/Ramsey's_Theorem#Proof_for_Two_Colors) for the two color case. There are still some ambiguities.

It says $R(r-1,s)$ and $R(r,s-1)$ exists by the inductive hypothesis. I know the principle of mathematical induction, but I still don't see it.

Furthermore it says in the proof that either $|M| \geq R(r-1,s)$ or $|N| \geq R(r,s-1)$. Why does this hold? I understand that $R(r-1,s) + R(r,s-1) -1 = |M| + |N|$.