Power-law Bounds for Increasing Subsequences in Brownian Separable Permutons and Homogeneous Sets in Brownian Cographons

The Brownian separable permutons are a one-parameter family -- indexed by p is an element of(0,1) -- of universal limits of random constrained permutations. We show that for each p is an element of(0,1), there are explicit constants 1/2infinity. In the symmetric case p=1/2, we have alpha & lowast;(p)approximate to 0.812 and beta & lowast;(p)approximate to 0.975. We present numerical simulations which suggest that the lower bound alpha & lowast;(p) is close to optimal in the whole range p is an element of(0,1). Our results work equally well for the closely related Brownian cographons. In this setting, we show that for each p is an element of(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 & lowast;(p)-o(1) )and n(beta & lowast;(p)+o(1)) (resp. n(alpha & lowast;(1-p)-o(1)) and n(beta & lowast;(1-p)+o(1))) with probability tending to 1 as n ->infinity. 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.