Covering n-Permutations with (n+1)-Permutations

Let Sn be the set of all permutations on [n] := {1, 2, . . . , n}. We denote by κn the smallest cardinality of a subset A of Sn+1 that “covers” Sn, in the sense that each π ∈ Sn may be found as an order-isomorphic subsequence of some π′ in A. What are general upper bounds on κn? If we randomly select νn elements of Sn+1, when does the probability that they cover Sn transition from 0 to 1? Can we provide a fine-magnification analysis that provides the “probability of coverage” when νn is around the level given by the phase transition? In this paper we answer these questions and raise others.