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(xy), ℙ(yx)}, 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 Tn of width 2 with δ(Tn) → 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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Correspondence to Ashwin Sah.

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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