The longest increasing subsequence of Brownian separable permutons

We establish a scaling limit result for the length LIS(σₙ) of the longest increasing subsequence of a permutation σₙ of size n sampled from the Brownian separable permuton μₚ of parameter p∈(0,1), which is the universal limit of pattern-avoiding permutations. Specifically, we prove that \[LIS(σ_n)/n^α\;{n→∞}{{a.s.}{}}\; X,\] where $α=α(p)$ is the unique solution in the interval $(1/2,1)$ to the equation \[{1}{41/2α√π}\,{Γ(1/2-1/2α)}{Γ(1-1/2α)}=p/p-1,\] and $X=X(p)$ is a non-deterministic and a.s. positive and finite random variable, which is a measurable function of the Brownian separable permuton. Notably, the exponent $α(p)$ is an increasing continuous function of p with α(0⁺)=1/2, α(1⁻)=1 and α(1/2)≈0.815226, which corresponds to the permuton limit of uniform separable permutations. We prove analogous results for the size of the largest clique of a graph sampled from the Brownian cographon of parameter p∈(0,1).