Improving the Lower Bound for the Union-closed Sets Conjecture via Conditionally IID Coupling (2306.08824v1)

Abstract: Recently, Gilmer proved the first constant lower bound for the union-closed sets conjecture via an information-theoretic argument. The heart of the argument is an entropic inequality involving the OR function of two i.i.d.\ binary vectors, and the best constant obtainable through the i.i.d.\ coupling is $\frac{3-\sqrt{5}}{2}\approx0.38197$. Sawin demonstrated that the bound can be strictly improved by considering a convex combination of the i.i.d.\ coupling and the max-entropy coupling, and the best constant obtainable through this approach is around 0.38234, as evaluated by Yu and Cambie. In this work we show analytically that the bound can be further strictly improved by considering another class of coupling under which the two binary sequences are i.i.d.\ conditioned on an auxiliary random variable. We also provide a new class of bounds in terms of finite-dimensional optimization. For a basic instance from this class, analysis assisted with numerically solved 9-dimensional optimization suggests that the optimizer assumes a certain structure. Under numerically verified hypotheses, the lower bound for the union-closed sets conjecture can be improved to approximately 0.38271, a number that can be defined as the solution to an analytic equation.