  One of the most classical results in Ramsey theory is the theorem of
Erd\H{o}s and Szekeres from 1935, which says that every sequence of more than
$k^2$ numbers contains a monotone subsequence of length $k+1$. We address the
following natural question motivated by this result: Given integers $k$ and $n$
with $n \geq k^2+1$, how many monotone subsequences of length $k+1$ must every
sequence of $n$ numbers contain? We answer this question precisely for all
sufficiently large $k$ and $n \leq k^2 + c k^{3/2} / \log k$, where $c$ is some
absolute positive constant.
