## 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*)*∈P*^{2} 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

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.

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Sah, A. Improving the \(\frac{1}{3} - \frac{2}{3}\) Conjecture for Width Two Posets.
*Combinatorica* **41, **99–126 (2021). https://doi.org/10.1007/s00493-020-4091-3

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### Mathematics Subject Classification (2010)

- 06A07
- 05A20
- 05D99