The proof of Ramsey's Theorem

Question

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 $R(r-1,s) + R(r,s-1) -1 = |M| + |N|$.

Answer 1

Here's something about the second question:

Suppose that $a,b,x,y$ are positive, and suppose that $$ a + b + 1 \leq x + y $$ Then either $x > a$ or $y > b$.

Proof: Suppose that we do not have $x > a$, so that $x \leq a$. We then note that $$ a + b + 1 - x = (a-x) + b + 1\leq y $$ That is, we have $$ y \geq (a - x) + b + 1 \geq b + 1 > b $$ So, we have $y > b$.

Answer 2

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.