Thresholds

Abstract:

Thresholds for increasing properties are a central concern in probabilistic combinatorics and elsewhere. (An increasing property, say F, is a superset-closed family of subsets of some [here finite] set X, and the “threshold question” for F asks, roughly, about how many random elements of X should one choose to make it likely that the resulting set lies in F?  For example:  about how many random edges from the complete graph on n vertices are typically required to produce a Hamiltonian cycle?)

We will try to give some sense of this area and then focus on a few recent highlights.