# Long Symmetric Chains in the Boolean Lattice

@article{Bajnok1996LongSC,
title={Long Symmetric Chains in the Boolean Lattice},
author={B{\'e}la Bajnok and Shahriar Shahriari},
journal={J. Comb. Theory, Ser. A},
year={1996},
volume={75},
pages={44-54}
}
• Published 1 July 1996
• Mathematics, Computer Science
• J. Comb. Theory, Ser. A
Let 2n]be the poset of all subsets of a set with n elements ordered by inclusion. A long chain in this poset is a chain ofn?1 subsets starting with a subset with one element and ending with a subset withn?1 elements. In this paper we prove: Given any collection of at mostn?2 skipless chains in 2n], there exists at least one (but sometimes not more than one) long chain disjoint from the chains in the collection. Furthermore, fork?3, given a collection ofn?kskipless chains in 2n], there are at…
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## References

SHOWING 1-10 OF 12 REFERENCES
Minimum cutsets for an element of a boolean lattice
• Mathematics
• 1989
An informative new proof is given for the theorem of Nowakowski that determines for all n and k the minimum size of a cutset for an element A with |A|=k of the Boolean algebra Bn of all subsets of
Cutsets of Boolean lattices
Derangements on the n-cube
• Mathematics
Discret. Math.
• 1993
Packing lines in a hypercube
• Mathematics
Discret. Math.
• 1993
A minimal cutset of the boolean lattice with almost all members
• Mathematics
Graphs Comb.
• 1989
Two almost explicit constructions are given satisfying the title of "almost explicit construction" of the type of graph derived from the inequality of the following type: graph of topographies.
Improved Lower Bounds on the Reliability of Hypercube Architectures
• Computer Science
IEEE Trans. Parallel Distributed Syst.
• 1994
Algorithms to compute lower bounds on TR andNR for the hypercube considering node and/or link failures are presented, which provide tighter bounds for both TR and NR than known results and run in time polynomial in the cube dimension n, specifically, within time O(n/sup 2/).
The Combinatorics of Network Reliability
The reliability polynominal Edge-disjoint subgraphs Additive and multiplicative improvements Combining the bounds The k-cycle bound Computational results References Index.