Breaking the Infinite Barrier in the $\frac{1}{3}$--$\frac{2}{3}$ Conjecture

The balance constant is a poset parameter measuring how well random linear extensions can be split according to their relative order on two elements. We give an $\epsilon$-improvement for the balance constant over the bound by Brightwell, Felsner and Trotter (1995), for some small $\epsilon >0$. This is the first general result towards the $\frac{1}{3}$–$\frac{2}{3}$ conjecture in over 30 years.