Let N be a finite set, let p∈(0,1), and let Np​ denote a random
binomial subset of N where every element of N is taken to belong to the
subset independently with probability p . This defines a product measure
μp​ on the power set of N, where for A⊆2Nμp​(A):=Pr[Np​∈A].
In this paper we study upward-closed families A for which all
minimal sets in A have size at most k, for some positive integer
k. We prove that for such a family μp​(A)/pk is a
decreasing function, which implies a uniform bound on the coarseness of the
thresholds of such families.
We also prove a structure theorem which enables to identify in A
either a substantial subfamily A0​ for which the first moment
method gives a good approximation of its measure, or a subfamily which can be
well approximated by a family with all minimal sets of size strictly smaller
than k.
Finally, we relate the (fractional) expectation threshold and the probability
threshold of such a family, using duality of linear programming. This is
related to the threshold conjecture of Kahn and Kalai