scholarly journals Sperner's Problem for G-Independent Families

2014 ◽  
Vol 24 (3) ◽  
pp. 528-550
Author(s):  
VICTOR FALGAS-RAVRY
Keyword(s):  
Maximal Antichain ◽  
Independent Edge ◽  
Sperner's Theorem ◽  
Independent Families ◽  
Free Sets

Given a graph G, let Q(G) denote the collection of all independent (edge-free) sets of vertices in G. We consider the problem of determining the size of a largest antichain in Q(G). When G is the edgeless graph, this problem is resolved by Sperner's theorem. In this paper, we focus on the case where G is the path of length n − 1, proving that the size of a maximal antichain is of the same order as the size of a largest layer of Q(G).

scholarly journals Non-meager free sets and independent families

2017 ◽  
Vol 145 (9) ◽  
pp. 4061-4073 ◽  
Author(s):  
Andrea Medini ◽  
Dušan Repovš ◽  
Lyubomyr Zdomskyy
Keyword(s):  
Independent Families ◽  
Free Sets

Characterization of the conditional stationary distribution in Markov chains via systems of linear inequalities

2020 ◽  
Vol 52 (4) ◽  
pp. 1249-1283
Author(s):  
Masatoshi Kimura ◽  
Tetsuya Takine
Keyword(s):  
Markov Chains ◽  
Stationary Distribution ◽  
Continuous Time ◽  
State Transition ◽  
Convex Polytope ◽  
Linear Inequalities ◽  
Systems Of Linear Inequalities ◽  
Free Sets ◽  
Practical Implications

AbstractThis paper considers ergodic, continuous-time Markov chains $\{X(t)\}_{t \in (\!-\infty,\infty)}$ on $\mathbb{Z}^+=\{0,1,\ldots\}$ . For an arbitrarily fixed $N \in \mathbb{Z}^+$ , we study the conditional stationary distribution $\boldsymbol{\pi}(N)$ given the Markov chain being in $\{0,1,\ldots,N\}$ . We first characterize $\boldsymbol{\pi}(N)$ via systems of linear inequalities and identify simplices that contain $\boldsymbol{\pi}(N)$ , by examining the $(N+1) \times (N+1)$ northwest corner block of the infinitesimal generator $\textbf{\textit{Q}}$ and the subset of the first $N+1$ states whose members are directly reachable from at least one state in $\{N+1,N+2,\ldots\}$ . These results are closely related to the augmented truncation approximation (ATA), and we provide some practical implications for the ATA. Next we consider an extension of the above results, using the $(K+1) \times (K+1)$ ( $K > N$ ) northwest corner block of $\textbf{\textit{Q}}$ and the subset of the first $K+1$ states whose members are directly reachable from at least one state in $\{K+1,K+2,\ldots\}$ . Furthermore, we introduce new state transition structures called (K, N)-skip-free sets, using which we obtain the minimum convex polytope that contains $\boldsymbol{\pi}(N)$ .

Simplicial elimination schemes, extremal lattices and maximal antichain lattices

Order ◽  
1996 ◽  
Vol 13 (2) ◽  
pp. 159-173 ◽  
Author(s):  
Michel Morvan ◽  
Lhouari Nourine
Keyword(s):  
Maximal Antichain

A novel 7·9 kb deletion causing α+ -thalassaemia in two independent families of Indian origin

2003 ◽  
Vol 120 (2) ◽  
pp. 364-366 ◽  
Author(s):  
Cornelis L. Harteveld ◽  
Peter Van Delft ◽  
Pierre W. Wijermans ◽  
Mies C. Kappers-Klunne ◽  
Jitske Weegenaar ◽  
...  
Keyword(s):  
Indian Origin ◽  
Independent Families

scholarly journals Strong versions of Sperner's theorem

1976 ◽  
Vol 20 (1) ◽  
pp. 80-88 ◽  
Author(s):  
Curtis Greene ◽  
Daniel J Kleitman
Keyword(s):  
Sperner's Theorem

On the Density of Universal Sum-Free Sets

1999 ◽  
Vol 8 (3) ◽  
pp. 277-280 ◽  
Author(s):  
TOMASZ SCHOEN
Keyword(s):  
Free Set ◽  
Free Sets

A set A is called universal sum-free if, for every finite 0–1 sequence χ = (e1, …, en), either(i) there exist i, j, where 1[les ]j<i[les ]n, such that ei = ej = 1 and i − j∈A, or(ii) there exists t∈N such that, for 1[les ]i[les ]n, we have t + i∈A if and only if ei = 1.It is proved that the density of each universal sum-free set is zero, which settles a problem of Cameron.

scholarly journals A generalization of Sperner's theorem

1981 ◽  
Vol 31 (4) ◽  
pp. 481-485 ◽  
Author(s):  
D. E. Daykin ◽  
P. Frankl ◽  
C. Greene ◽  
A. J. W. Hilton
Keyword(s):  
Sperner's Theorem ◽  
Lym Inequality ◽  
Families Of Subsets

AbstractSome generalizations of Sperner's theorem and of the LYM inequality are given to the case when A1,… At are t families of subsets of {1,…,m} such that a set in one family does not properly contain a set in another.

Sum-free sets and short covering codes

2010 ◽  
Vol 39 (6) ◽  
Author(s):  
Emerson Carmelo ◽  
Candido Mendonça
Keyword(s):  
Covering Codes ◽  
Short Covering ◽  
Free Sets
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