Beyond Liu's 0.382709 threshold for the union-closed sets conjecture

The union-closed sets conjecture asks whether every finite union-closed family containing a nonempty set has an element contained in at least half of its members. Let $c_{\mathrm{U}\mathrm{C}}$ denote the largest universal lower bound on this proportion. The best previously proved lower bound is due to Liu and is above $0.3823455$; his own numerical optimization suggested a stronger threshold $c_{\mathrm{L}}>0.382709087918735$. The main argument of this paper is to use different protocols (suitable families of probability measures) within Liu’s framework. In particular, we combine the independent protocol with Liu’s Example 5 to prove the conjectured bound $c_{\mathrm{U}\mathrm{C}}\ge c_{\mathrm{L}}$. We then vary the parameter in Example 5 and optimize the choice of the two coefficients in the combination to obtain $c_{\mathrm{U}\mathrm{C}}\ge 0.3828852549667978$. We also show that the inequality used in this proof cannot give a threshold larger than $0.3828852599667\dots$.