Uniform separable permutations and uniform cographs

Prove the convergence in distribution for the longest increasing subsequence of a uniform separable permutation and for the largest independent set of a uniform separable cograph, with the common exponent α(1/2), limiting variable X(1/2), and deterministic multiplicative constant c>0 specified in Conjecture 1.1.

Background

The paper establishes scaling limits for permutations and graphs sampled directly from the Brownian separable permuton and Brownian cographon. Uniform separable permutations and uniform separable cographs have the same continuum limits at parameter 1/2, but their discrete sampling mechanisms differ.

The authors conjecture that the longest increasing subsequence and largest independent-set observables nevertheless have the same exponent and limiting random-variable behavior, up to a deterministic constant c.

References

We believe that our results can be extended to the length of the longest increasing subsequence in uniform separable permutations and the size of the largest clique and independent set in uniform cographs (and, more generally, to most natural models known to converge to the Brownian separable permuton or the Brownian cographon).

— The longest increasing subsequence of Brownian separable permutons  (2506.19123 - Adhikari et al., 23 Jun 2025) in Conjecture 1.1, Section 1.3, “Some conjectures and additional comments”