Let $m,n$ be natural numbers. Prove using Ramsey's theorem there's a minimal natural $N$ for which every sequence of length $N$ contains either an increasing subsequence of length $m+1$ or a decreasing subsequence of order $n+1$. What's the minimal value of $N$ for which the claim holds? What's the upper bound given by Ramsey's theorem?
$R(m+1,n+1)$ is the minimal such $N$ and it exists by Ramsey theorem - simply color an edge $ \left\{ i,j \right\}$ blue if $a_i\leq a_j$ and red otherwise. The upper bound given is also $\binom{m+n-4}{m}$. Thing is, I don't see how to calculate the minimal such $N$ using Ramsey's theorem. Am I supposed to prove Erdős–Szekeres for this small part and then prove its bound is tight, or am I missing something?