Improving the $\frac{1}{3}-\frac{2}{3}$ Conjecture for Width Two Posets

Extending results of Linial (1984) and Aigner (1985), we prove a uniform lower bound on the balance constant of a poset $P$ of width $2$. This constant is defined as $\delta(P) = \max_{(x, y)\in P^2}\min\{\mathbb{P}(x\prec y), \mathbb{P}(y\prec x)\}$, where $\mathbb{P}(x\prec y)$ is the probability $x$ is less than $y$ in a uniformly random linear extension of $P$. In particular, we show that if $P$ is a width $2$ poset that cannot be formed from the singleton poset and the three element poset with one relation using the operation of direct sum, then $$\delta(P)\ge\frac{-3 + 5\sqrt{17}}{52}\approx 0.33876\ldots.$$ This partially answers a question of Brightwell (1999); a full resolution would require a proof of the $\frac{1}{3}-\frac{2}{3}$ Conjecture that if $P$ is not totally ordered then $\delta(P)\ge\frac{1}{3}$. Furthermore, we construct a sequence of posets $T_n$ of width $2$ with $\delta(T_n)\rightarrow\beta\approx 0.348843\ldots$, giving an improvement over a construction of Chen (2017) and over the finite posets found by Peczarski (2017). Numerical work on small posets by Peczarski suggests the constant $\beta$ may be optimal.