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Colorful partitions of cardinal numbers

J. Baumgartner, P. Erdős, F. Galvin, J. Larson
1979 Canadian Journal of Mathematics - Journal Canadien de Mathematiques  
The letters K, X, p, Y, k, n, are reserved for cardinal numbers, while 01, 0, y, 6, n, b are used for ordinals. Each ordinal number is identified with the set of its predecessors.  ...  N ow S induces a partition of X = U(B,:a <cfp},whereB, = (y:S(y) =ai.Foreachcr<cfp,Rrestrictedto X X & maps into A,.  ... 
doi:10.4153/cjm-1979-056-4 fatcat:p352iwuh6rbepinintqvqav7te

Cardinal numbers with partition properties

Neil H. Williams
1970 Bulletin of the Australian Mathematical Society  
A number of important open questions remain, however. The final chapter, Chapter V, is unrelated to the earlier ones. This is concerned with a development of the forcing method of P.J.  ...  Chapters II and IV are concerned with the generalization of the relation (l), a generalization obtained by considering not partitions of the n-element subsets of K , but rather partitions of certain finite  ...  Cardinal numbers with partition properties Neil H.  ... 
doi:10.1017/s0004972700045779 fatcat:atvz2hzc4baaxgt2d5mwwh74eq

The b-domatic number of a graph

Odile Favaron
2013 Discussiones Mathematicae Graph Theory  
A partition P of G is a partition of its vertex set V . Its cardinality |P| is the number of classes of P.  ...  We introduce the b-domatic number bd(G) as the counterpart of the b-chromatic number by giving an alternative definition of the maximality of a partition into dominating sets.  ...  The authors of [10] solved the interpolation problem for the a-chromatic number by proving that every graph G admits a-minimal chromatic partitions of any  ... 
doi:10.7151/dmgt.1709 fatcat:tb3cgqvvo5fjho46zponoq65da

An order property of partition cardinals

N. H. Williams
1970 Bulletin of the Australian Mathematical Society  
This note studies cardinal numbers K which have a partition property which amounts to the following. Let V be a cardinal, H an ordinal limit number and m a positive integer.  ...  The set of all finite subsets of S is denoted [ S ] < u . Cardinal numbers are identified with the initial ordinals.  ...  Let K and v be cardinals, let n be an ordinal limit number and let m be a positive integer. Suppose that for any partition A = {A, ; I < v} of K into v parts the following situation prevails.  ... 
doi:10.1017/s0004972700045834 fatcat:fyijunszyzcdtgcd5zmhjgfbbe

Dualization of the van Douwen diagram

Jacek Cichoń, Adam Krawczyk, Barbara Majcher-Iwanow, Bogdan Wȩglorz
2000 Journal of Symbolic Logic (JSL)  
AbstractWe make a more systematic study of the van Douwen diagram for cardinal coefficients related to combinatorial properties of partitions of natural numbers.  ...  With any partition X e (co) we can consider some finite modification of X, denoted by X* 9 which is a partition obtained from X by gluing together a finite number of pieces of X.  ...  coefficient t, i.e. the minimal cardinal number being the cardinality of a family well-ordered by the inverse almost containedness without lower bounds.  ... 
doi:10.2307/2586580 fatcat:5g7r2ioxkbgjlp3zemne6gsxlu

Crossings and nestings in set partitions of classical types

Martin Rubey, Christian Stump
2010 Discrete Mathematics & Theoretical Computer Science  
On the other hand we generalize a bijection to type $B$ and $C$ that interchanges the cardinality of a maximal crossing with the cardinality of a maximal nesting, as given by Chen, Deng, Du, Stanley and  ...  International audience In this extended abstract, we investigate bijections on various classes of set partitions of classical types that preserve openers and closers.  ...  Stanley [4] have shown that the number of set partitions where a maximal crossing has cardinality k and a maximal nesting has cardinality is the same as the number of set partitions where a maximal crossing  ... 
doi:10.46298/dmtcs.2826 fatcat:3dfg4xzokffetgxaixgf5nuhou

On shattering, splitting and reaping partitions [article]

Lorenz Halbeisen
2001 arXiv   pre-print
Using some properties of the ideal J of nowhere dual-Ramsey sets, which is an ideal over the set of partitions of omega, we show that add(J)=cov(J)=H.  ...  In this article we investigate the dual-shattering cardinal H, the dual-splitting cardinal S and the dual-reaping cardinal R, which are dualizations of the well-known cardinals h (the shattering cardinal  ...  The cardinal number wS is the least cardinal number κ, for which there exists a weak splitting family of cardinality κ. (It is obvious that wS ≤ S ′ .)  ... 
arXiv:math/0109099v1 fatcat:ekkttasrujcjhe4a5pnogybhay

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

James Emil Avery, Jean-Yves Moyen, Pavel Ruzicka, Jakob Grue Simonsen
2017 arXiv   pre-print
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  ...  Moreover we can construct maximal antichains of cardinality (κ, 2^λ) for any λ<κ; (V) all cardinals of the form κ^λ with 0 <λ<κ occur as the number of complements to some partition P∈Π_κ, and only these  ...  James Avery was supported by VILLUM FONDEN through the network for Experimental Mathematics in Number Theory, Operator Algebras, and Topology.  ... 
arXiv:1501.05284v4 fatcat:rqkjpehqxra63efgpujgsre57i

Page 2276 of Mathematical Reviews Vol. , Issue 83f [page]

1983 Mathematical Reviews  
of cardinality « such that all the increasing sequences of elements of C lie in the same part of the partition.  ...  power set of every ordinal exists, and there are arbitrarily large cardinals with the strong partition property.  ... 

Using partitions to characterize the minimum cardinality of an unbounded family in ωω

L.Brian Lawrence
1990 Journal of combinatorial theory. Series A  
Suppose v is an infinite cardinal. We use the term "partition" to refer to a partition of v with a countably intinite number of cells.  ...  Cardinal numbers. With respect to "CO partially ordered by < *, let b be the minimum cardinality of an unbounded family, and let a' be the minimum cardinality of a dominant (colinal) family.  ... 
doi:10.1016/0097-3165(90)90027-t fatcat:lnkbb3d7rbb5new4pkder2lizi

Weak partition properties for infinite cardinals. I

E. M. Kleinberg
1971 Proceedings of the American Mathematical Society  
Partition properties are perhaps the most fruitful of the various methods for defining and discussing large cardinals in set theory.  ...  In this paper we weaken in a natural way the most well known of these partition properties and examine the extent to which the cardinals defined remain "large."  ...  An area of set theory which has come under a great deal of study recently is that concerned with partition properties for cardinal numbers.  ... 
doi:10.1090/s0002-9939-1971-0281626-7 fatcat:rgavodpzznew3ipqu5wusobbee

On partitions of discrete boxes

Noga Alon, Tom Bohman, Ron Holzman, Daniel J. Kleitman
2002 Discrete Mathematics  
We prove that any partition of an n-dimensional discrete box into nontrivial sub-boxes must consist of at least 2 n sub-boxes, and consider some extensions of this theorem.  ...  We are also grateful to Jeff Kahn for suggesting the point of view taken in subsection 2.4.  ...  A sub-box is odd if and only if each of its n factors has odd cardinality, and the nontriviality of the B j implies that half of the odd cardinality subsets of A i intersect B j i in an odd number of elements  ... 
doi:10.1016/s0012-365x(02)00428-4 fatcat:gsi3xpnyhnfthcf5lqhqcn5spq

Weak Partition Properties for Infinite Cardinals. I

E. M. Kleinberg
1971 Proceedings of the American Mathematical Society  
Partition properties are perhaps the most fruitful of the various methods for defining and discussing large cardinals in set theory.  ...  In this paper we weaken in a natural way the most well known of these partition properties and examine the extent to which the cardinals defined remain "large."  ...  An area of set theory which has come under a great deal of study recently is that concerned with partition properties for cardinal numbers.  ... 
doi:10.2307/2038284 fatcat:3zbzqxb2urb7bl3eurvugliozu

A complete anytime algorithm for balanced number partitioning [article]

Stephan Mertens
1999 arXiv   pre-print
the cardinalities of the subsets be within one of each other.  ...  Given a set of numbers, the balanced partioning problem is to divide them into two subsets, so that the sum of the numbers in each subset are as nearly equal as possible, subject to the constraint that  ...  The numbers in small font are the effective cardinalities needed to keep track of the cardinality difference of the final partition. The dashed parts of the tree are pruned by the algorithm.  ... 
arXiv:cs/9903011v1 fatcat:iuuajo6yfrhuzfjhwul4xxpwnq

Cached Iterative Weakening for Optimal Multi-Way Number Partitioning

Ethan Schreiber, Richard Korf
2014 PROCEEDINGS OF THE THIRTIETH AAAI CONFERENCE ON ARTIFICIAL INTELLIGENCE AND THE TWENTY-EIGHTH INNOVATIVE APPLICATIONS OF ARTIFICIAL INTELLIGENCE CONFERENCE  
The NP-hard number-partitioning problem is to separate a multiset S of n positive integers into k subsets, such that the largest sum of the integers assigned to any subset is minimized.  ...  instead of at each node of the search tree; and explores subsets in cardinality order instead of an arbitrary order.  ...  While number partitioning fixes the number of subsets k and minimizes the sum of the largest subset, bin packing fixes the maximum sum of the subsets (bins) and minimizes the number of subsets needed.  ... 
doi:10.1609/aaai.v28i1.9122 fatcat:34qwpmrskbenrjzijzuxn2d52m
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