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

Abstract

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 δ(P) = max(x,y)∈P² min{ℙ(x ≺ y), ℙ(y ≺ x)}, where ℙ(x≺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

$$\delta \left( P \right) \ge \frac{{ - 3 + 5\sqrt {17} }}{{52}} \approx 0.33876....$$

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 $\delta \left( P \right) \ge \frac{1}{3}$.

Furthermore, we construct a sequence of posets T_n of width 2 with δ(T_n) → β ≈ 0.348843…, 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 β may be optimal.