scholarly journals Note on a Ramsey Theorem for Posets with Linear Extensions

10.37236/6392 ◽  
2017 ◽  
Vol 24 (4) ◽  
Author(s):  
Andrii Arman ◽  
Vojtěch Rödl
Keyword(s):  
Short Proof ◽  
Partially Ordered Sets ◽  
Linear Extensions ◽  
Ordered Sets ◽  
Type Result ◽  
Ramsey Theorem ◽  
Partially Ordered ◽  
Ramsey Type

In this note we consider a Ramsey-type result for partially ordered sets. In particular, we give an alternative short proof of a theorem for a posets with multiple linear extensions recently obtained by Solecki and Zhao (2017).

scholarly journals A Family of Partially Ordered Sets with Small Balance Constant

10.37236/7337 ◽  
2018 ◽  
Vol 25 (4) ◽  
Author(s):  
Evan Chen
Keyword(s):  
Partially Ordered Sets ◽  
Linear Extensions ◽  
Ordered Sets ◽  
Partially Ordered ◽  
Finite Poset ◽  
A Chain

Given a finite poset $\mathcal P$ and two distinct elements $x$ and $y$, we let $\operatorname{pr}_{\mathcal P}(x \prec y)$ denote the fraction of linear extensions of $\mathcal P$ in which $x$ precedes $y$. The balance constant $\delta(\mathcal P)$ of $\mathcal P$ is then defined by \[ \delta(\mathcal P) = \max_{x \neq y \in \mathcal P} \min \left\{ \operatorname{pr}_{\mathcal P}(x \prec y), \operatorname{pr}_{\mathcal P}(y \prec x) \right\}. \] The $1/3$-$2/3$ conjecture asserts that $\delta(\mathcal P) \ge \frac13$ whenever $\mathcal P$ is not a chain, but except from certain trivial examples it is not known when equality occurs, or even if balance constants can approach $1/3$.In this paper we make some progress on the conjecture by exhibiting a sequence of posets with balance constants approaching $\frac{1}{32}(93-\sqrt{6697}) \approx 0.3488999$, answering a question of Brightwell. These provide smaller balance constants than any other known nontrivial family.

On Operations and Linear Extensions of Well Partially Ordered Sets

Order ◽  
2004 ◽  
Vol 21 (1) ◽  
pp. 7-17 ◽  
Author(s):  
Maciej Malicki ◽  
Aleksander Rutkowski
Keyword(s):  
Partially Ordered Sets ◽  
Linear Extensions ◽  
Ordered Sets ◽  
Partially Ordered

scholarly journals Pattern avoidance in forests of binary shrubs

2016 ◽  
Vol Vol. 18 no. 2, Permutation... (Permutation Patterns) ◽  
Author(s):  
David Bevan ◽  
Derek Levin ◽  
Peter Nugent ◽  
Jay Pantone ◽  
Lara Pudwell ◽  
...  
Keyword(s):  
Growth Rate ◽  
Generating Function ◽  
Minimal Polynomial ◽  
Partially Ordered Sets ◽  
Pattern Avoidance ◽  
Lattice Paths ◽  
Linear Extensions ◽  
Ordered Sets ◽  
Analytic Combinatorics ◽  
Partially Ordered

We investigate pattern avoidance in permutations satisfying some additional restrictions. These are naturally considered in terms of avoiding patterns in linear extensions of certain forest-like partially ordered sets, which we call binary shrub forests. In this context, we enumerate forests avoiding patterns of length three. In four of the five non-equivalent cases, we present explicit enumerations by exhibiting bijections with certain lattice paths bounded above by the line $y=\ell x$, for some $\ell\in\mathbb{Q}^+$, one of these being the celebrated Duchon's club paths with $\ell=2/3$. In the remaining case, we use the machinery of analytic combinatorics to determine the minimal polynomial of its generating function, and deduce its growth rate.

To the spectral theory of partially ordered sets

2018 ◽  
Vol 60 (3) ◽  
pp. 578-598
Author(s):  
Yu. L. Ershov ◽  
M. V. Schwidefsky
Keyword(s):  
Spectral Theory ◽  
Partially Ordered Sets ◽  
Ordered Sets ◽  
Partially Ordered
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