The length of the longest increasing subsequence in a random permutation and the size of the largest homogeneous set (i.e. a clique or an independent set) in a random graph are two of the classical problems at the interface of combinatorics and probability theory, with connections to several other areas of mathematics.
In this paper, we investigated these classical problems in the setting of universal Brownian-type permutations and graphs, i.e. for the Brownian separable permutons and the Brownian cographons. These objects are the universal limits of various random permutations and graph families.