  The Brownian separable permutons are a one-parameter family -- indexed by
$p\in(0,1)$ -- of universal limits of random constrained permutations. We show
that for each $p\in (0,1)$, there are explicit constants $1/2 < \alpha_*(p)
\leq \beta^*(p) < 1$ such that the length of the longest increasing subsequence
in a random permutation of size $n$ sampled from the Brownian separable
permuton is between $n^{\alpha_*(p) - o(1)}$ and $n^{\beta^*(p) + o(1)}$ with
probability tending to 1 as $n\to\infty$. In the symmetric case $p=1/2$, we
have $\alpha_*(p) \approx 0.812$ and $\beta^*(p)\approx 0.975$. We present
numerical simulations which suggest that the lower bound $\alpha_*(p)$ is close
to optimal in the whole range $p\in(0,1)$.
  Our results work equally well for the closely related Brownian cographons. In
this setting, we show that for each $p\in (0,1)$, the size of the largest
clique (resp. independent set) in a random graph on $n$ vertices sampled from
the Brownian cographon is between $n^{\alpha_*(p) - o(1)}$ and $n^{\beta^*(p) +
o(1)}$ (resp. $n^{\alpha_*(1-p) - o(1)}$ and $n^{\beta^*(1-p) + o(1)}$) with
probability tending to 1 as $n\to\infty$.
  Our proofs are based on the analysis of a fragmentation process embedded in a
Brownian excursion introduced by Bertoin (2002). We expect that our techniques
can be extended to prove similar bounds for uniform separable permutations and
uniform cographs.
