Improving the 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 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 ext...

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Veröffentlicht in:Combinatorica (Budapest. 1981) 2021, Vol.41 (1), p.99-126
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description 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 Conjecture that if P is not totally ordered, then . 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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Lower bounds
Mathematics
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Original Paper
Set theory
title Improving the Conjecture for Width Two Posets
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