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On maximal antichains containing no set and its complement

G.F. Clements, H.-D.O.F. Gronau
1981 Discrete Mathematics  
J'e:Tolci Griggs and the referee for some fruitful wggestions. On rnaxi~tnal antichains con aining no set 1 ml its cmnpkment  ...  of S if ;And only if X contains 1.:~ ;tctly one vector from each pair.  ...  or its complement:(i) if n is odd!  ... 
doi:10.1016/0012-365x(81)90267-3 fatcat:eimeprubnfaxxfhakcttfhfxdy

A splitting property of maximal antichains

Rudolf Ahlswede, Péter L. Erdős, Niall Graham
1995 Combinatorica  
In any dense poset P (e.g. in the Boolean lattice) every maximal antichain S may be partitioned into disjoint subsets S 1 and S 2 , such that the union of the downset of S 1 with the upset of S 2 yields  ...  To find a similar splitting of maximal antichains in posets is NP-hard in general.  ...  It is easy to find a poset P and a maximal antichain S in P such that S has no splitting.  ... 
doi:10.1007/bf01192520 fatcat:mr547nzlrja7xgpdkxqh7omo4m

Chains, Antichains, and Complements in Infinite Partition Lattices [article]

James Emil Avery, Jean-Yves Moyen, Pavel Ruzicka, Jakob Grue Simonsen
2017 arXiv   pre-print
We consider the partition lattice Π_κ on any set of transfinite cardinality κ and properties of Π_κ whose analogues do not hold for finite cardinalities.  ...  Moreover, we give a direct formula for the number of complements to a given partition; (VI) Under the Generalized Continuum Hypothesis, the cardinalities of maximal chains, maximal antichains, and numbers  ...  The authors thank Benoît Kloeckner for his many comments on a preliminary version of this article, and thank the anonymous referees for valuable suggestions.  ... 
arXiv:1501.05284v4 fatcat:rqkjpehqxra63efgpujgsre57i

Note on maximal split-stable subgraphs

Ulrich Faigle, Bernhard Fuchs, Britta Peis
2007 Discrete Applied Mathematics  
In this note, we address the corresponding optimization problem: identify a set of vertices of maximal cardinality that decomposes into a red and a blue stable set.  ...  We investigate the complexity of special and related cases (e.g., (anti-)chains in partial orders and stable matroid bases) and exhibit some NP-hard cases as well as polynomial ones.  ...  Acknowledgments We thank anonymous referees on an earlier version of this paper for pointing out F. Gavril's work [6] , and Gyula Pap for fruitful conversations.  ... 
doi:10.1016/j.dam.2007.04.009 fatcat:gxgqqq5tkzc3rnnfghmeknv654

Chains, antichains, and fibres

Zbigniew Lonc, Ivan Rival
1987 Journal of combinatorial theory. Series A  
Call a subset of an ordered set a fibre if it meets every maximal antichain.  ...  One example is this: In a finite graph G every minimal transuersal is independent if and only if G contains no path of length three. !D 1987 Academic ems, hc.  ...  An ordered set is N-free if its diagram contains no N, that is, if it contains no cover-preserving subset isomorphic to (a < c, b < c, b <d).  ... 
doi:10.1016/0097-3165(87)90029-x fatcat:hpiplvqskzbnlcawu37w3pu4vq

On the size of maximal chains and the number of pairwise disjoint maximal antichains

Dwight Duffus, Bill Sands
2010 Discrete Mathematics  
in I k , then P contains k pairwise disjoint maximal antichains.  ...  For each integer k ≥ 3, we find all maximal intervals I k of natural numbers with the following property: whenever the number of elements in every maximal chain in a finite partially ordered set P lies  ...  Acknowledgements The research was largely performed while the second author was visiting Emory University on sabbatical. The second author thanks Emory University for its hospitality.  ... 
doi:10.1016/j.disc.2010.06.035 fatcat:4uskksdx3nc2rc77sqmfev4oiy

On Deductive Systems of AC Semantics for Rough Sets [article]

A. Mani
2016 arXiv   pre-print
In her paper two different semantics, one for general rough sets and another for general approximation spaces over quasi-equivalence relations, were developed.  ...  Antichain based semantics for general rough sets were introduced recently by the present author.  ...  Suppose the objects represented by the pairs (a, b) and (e, f ) are such that a = e and b ⊂ f , then it is clear that any maximal antichain containing (e, f ) cannot contain any element from {x : x l  ... 
arXiv:1610.02634v1 fatcat:rybwgw33azdgdhdp5jracsxuaa

Two-colouring all two-element maximal antichains

D. Duffus, B. Sands, N. Sauer, R.E. Woodrow
1991 Journal of combinatorial theory. Series A  
We give an example of a finite poset P with no one-element maximal antichain and containing no fibre of size at most JP(/2, thus answering a question of Aigner-Andreae and disproving a conjecture of Lone-Rival  ...  A fibre in a partially ordered set P is a subset of P meeting every maximal antichain of P.  ...  In this case only one kind of hypergraph (maximal complete subgraph, or maximal independent set) need be considered, since the complement of a perfect graph is perfect.  ... 
doi:10.1016/0097-3165(91)90009-6 fatcat:26fhjtvp5nbifbhexsj2jcmn7a

On maximal antichains consisting of sets and their complements

Hans-Dietrich O.F Gronau
1980 Journal of combinatorial theory. Series A  
The integers pi= I(i)XI, i =O, l,..., K, are called parameters of X Clements and Gronau 141 studied the maximal antichains containing no set and its complement.  ...  The opposite concept, c-antichains, i.e., antichains consisting of sets and their complements, was considered in the case k, = I by Bollob&s [ 11.  ... 
doi:10.1016/0097-3165(80)90034-5 fatcat:ar7wlqlqc5fotcymc7bgaaqeou

Covering Graphs by Colored Stable Sets

Ulrich Faigle, Bernhard Fuchs, Britta Wienand
2004 Electronic Notes in Discrete Mathematics  
Let G = (V, R ∪ B) be a multigraph with red and blue edges. G is an R/B-split graph if V is the union of a red and a blue stable set.  ...  On the other hand, finding a maximal R/B-subgraph is N P-hard already for the class of comparability graphs of series-parallel orders.  ...  Acknowledgment The authors are grateful to Nguyen Tien Thanh for pointing out the relationship between R/B-split graphs and König graphs.  ... 
doi:10.1016/j.endm.2004.03.029 fatcat:s3nhdkbwcbgm3nrzuzrwt7zy5e

A note on blockers in posets [article]

Anders Björner, Axel Hultman
2004 arXiv   pre-print
The blocker A^* of an antichain A in a finite poset P is the set of elements minimal with the property of having with each member of A a common predecessor. The following is done: 1.  ...  Connections with the question of finding minimal size blocking sets for certain set families are discussed.  ...  A crucial property in this setting is that if A is an antichain (no set contains another), then A * * = A.  ... 
arXiv:math/0403094v1 fatcat:igiqjkk2sfbjrjqwgjfiqi64ru

Page 4982 of Mathematical Reviews Vol. , Issue 86k [page]

1986 Mathematical Reviews  
Then (1) P is the union of antichain cutsets if and only if it contains no alternating-cover cycle; (2) z in P is contained in an antichain cutset if and only if there is no generalized alternating-cover  ...  Order 1 (1985), no. 3, 235-247. A subset A of an ordered set P is called a cutset if each maximal chain of P meets A. If, in addition, A is an antichain, it is called an antichain cutset.  ... 

Fibres and ordered set coloring

Dwight Duffus, H.A. Kierstead, W.T. Trotter
1991 Journal of combinatorial theory. Series A  
A Jbre F of a partially ordered set P is a subset which intersects each nontrivial maximal antichain of P.  ...  Let I be the smallest constant such that each finite partially ordered set P contains a fibre of size at most I '1 PI.  ...  Conversely, a maximal antichain A disjoint from F contains exactly one of the literals U, ii for each u E U.  ... 
doi:10.1016/0097-3165(91)90083-s fatcat:eimtxh5vt5dabnkxedwhr6tmoq

On a new reciprocity, distribution and duality law

G. Kurepa
1957 Pacific Journal of Mathematics  
All depend on the interconnection between maximal chains and maximal antichains of ordered sets. By considering ordered sets one achieves considerable generality.  ...  On the other hand, the parallel considerations of maximal chains and maximal antichains of S give rise to a new kind of interconnection of elements of P 1 !  ...  Proof Suppose, on the contrary, that T contains a maximal chain M and a maximal antichain A so that (10.3) A Γ\ M^v .  ... 
doi:10.2140/pjm.1957.7.1125 fatcat:4z6dxcuwpvfp3gttysuvxrcyk4

Two forbidden induced subgraphs and well-quasi-ordering

Nicholas Korpelainen, Vadim Lozin
2011 Discrete Mathematics  
Since two graphs G and H are isomorphic if and only if their complements are isomorphic, we conclude that Claim 2. A set of graphs is a WQO if and only if the set of their complements is a WQO.  ...  It is known that a class of graphs defined by a single forbidden induced subgraph G is well-quasi-ordered by the induced subgraph relation if and only if G is an induced subgraph of P 4 .  ...  Introduction A partial order on graphs is a well-quasi-order (WQO for short) if it contains no infinite antichains, i.e. no infinite sets of graphs pairwise incomparable with respect to the order.  ... 
doi:10.1016/j.disc.2011.04.023 fatcat:bq6wz72lgrfrvpth4xkatbi5am
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