  We study the problem of finding monotone subsequences in an array from the
viewpoint of sublinear algorithms. For fixed $k \in \mathbb{N}$ and
$\varepsilon > 0$, we show that the non-adaptive query complexity of finding a
length-$k$ monotone subsequence of $f \colon [n] \to \mathbb{R}$, assuming that
$f$ is $\varepsilon$-far from free of such subsequences, is $\Theta((\log
n)^{\lfloor \log_2 k \rfloor})$. Prior to our work, the best algorithm for this
problem, due to Newman, Rabinovich, Rajendraprasad, and Sohler (2017), made
$(\log n)^{O(k^2)}$ non-adaptive queries; and the only lower bound known, of
$\Omega(\log n)$ queries for the case $k = 2$, followed from that on testing
monotonicity due to Erg\"un, Kannan, Kumar, Rubinfeld, and Viswanathan (2000)
and Fischer (2004).
