The induction step in this proof of Ramsey's theorem states that
First we prove the theorem for the $2$-colour case, by induction on $r + s$. It is clear from the definition that for all $n$, $R(n, 1) = R(1, n) = 1$. This starts the induction. We prove that $R(r, s)$ exists by finding an explicit bound for it. By the inductive hypothesis $R(r − 1, s)$ and $R(r, s − 1)$ exist.
I am not exactly sure why the inductive requires that the statement is true for all $n$. If it works for $R(n,1)$, wouldn't that be enough since we can easily verify that the statement is true for $r+s=2$, and induct from there?