Linear-sized independent sets in random cographs and increasing subsequences in separable permutations

This paper is interested in independent sets (or equivalently, cliques) uniform random cographs. We also study their permutation analogs, namely, increasing subsequences separable permutations. First, we prove that, with high probability as \(n\) gets large, the largest set a cograph vertices has size \(o(n)\). answers question of Kang, McDiarmid, Reed and Scott. Using connection between graphs permutations via inversion graphs, give similar result for longest subsequence These results are proved using self-similarity Brownian limits cographs permutations, actually apply more generally to all families same limit. Second, unexpectedly given above results, show that \(\beta >0\) sufficiently small, expected number n\) grows exponentially fast \(n\). analog this result. time proofs rely on singularity analysis associated bivariate generating functions.Mathematics Subject Classifications: 60C05, 05C80, 05C69, 05A05Keywords: Combinatorial graph theory, combinatorial probability, cographs, graphons,