In this talk, I will introduce the new notion of the k-threshold of an increasing family, which generalizes the well-studied threshold function. I will show that Talagrand's discrete convexity conjecture is equivalent to the assertion that for some universal integer k, the k-threshold of every increasing family is at most a universal constant times its expectation threshold.

This talk will focus on graph properties: we can bound the k-threshold of any increasing graph family in terms of ordinary thresholds of graphs obtained by decomposing members of the family. We use this to show that the reformulated conjecture holds for several classical spanning graph containment properties, such as containment of perfect matchings and Hamiltonian cycles, and more generally for all graph containment properties whose target graphs have sufficiently small degeneracy. I'll end the talk with some tantalizing open problems. Joint work with Xiaoyu He, Jinyoung Park, and Michel Talagrand.