Reserve coupling for spread measures and independent noise

Establish that there is an absolute constant C such that, whenever B is an a-spread random subset of a finite set X with |B|≤r, X_p is independent of B, and p≥Ca log(2r), the law of B∪X_p admits a coupling with the product measure μ_{min{1,p+Ca}} under which B∪X_p is contained in the product random set with probability at least 3/4.

Background

The paper’s decomposition of a random copy of H into conditionally spread blocks produces a logarithmic number of spread components. The proposed coupling would absorb each component into independent product noise while increasing the product density by only O(a) per component.

The conjecture is designed to imply both the fractional Kahn–Kalai threshold theorem and the second Kahn–Kalai conjecture. The paper verifies it for several special classes of laws, including rank-one laws, independent unions of rank-one laws on disjoint blocks, strongly Rayleigh laws, and laws with log-concave multiaffine generating polynomials, but leaves the general statement unresolved.