Determine whether global-coupling entropy gains remain large for low-frequency families

Determine whether the entropy gain produced by the global inverse-multiplicity coupling can remain comparable to \(\ln N\) for union-closed families whose maximum element frequency is small.

Background

The paper combines a pointwise entropy argument with a global coupling of pairs of sets. The global coupling is obtained by minimizing relative entropy subject to uniform marginals, and it yields a strictly positive entropy improvement over independent sampling for every finite family size $N$. This produces a frequency bound of the form $p\ge c_0+\text{correction}(N)$.

The available size-only correction tends to zero as $N\to\infty$, so it does not improve the universal constant $c_0=0.38288525$. The authors identify the unresolved issue as whether the entropy gain from the global coupling can remain of order $\ln N$ when the family has small maximum element frequency; resolving this could potentially yield a stronger absolute frequency bound.