The proof of Ramsey's Theorem I try to understand the proof of Ramsey's Theorem 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 $R(r-1,s) + R(r,s-1) -1 = |M| + |N|$.

Let me try to answer the first question.
The inductive hypothesis is $R(r,s)$ exists.
We know $\forall n\in N, R(n,1)=R(1,n)=1$.
Assume $\forall r<r_0, s<s_0$, $R(r,s)$ exists. (induction hypothesis)
Then we want to show $R(r_0,s_0)$ exists.
Then we apply the "Proof for Two Colors" to show that $R(r_0,s_0)≤R(r_0−1,s_0)+R(r_0,s_0−1)$, which implies $R(r_0,s_0)$ exists.
