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Partitioning into Sets of Bounded Cardinality
[chapter]
2009
Lecture Notes in Computer Science
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We show that the partitions of an n-element set into k members of a given set family can be counted in time O((2− ) n ), where > 0 depends only on the maximum size among the members of the family. ...
Lemma 1 . 1 The number of partitions of N into k members of F equals the cardinality of L k .
Lemma 3 . 3 Let n and r be natural numbers. ...
In this paper, we answer the question affirmatively in the special case where the given set family consists of sets whose cardinality is bounded by a constant. ...
doi:10.1007/978-3-642-11269-0_21
fatcat:abr6k7xxsncepfcdtnl4bplkau
A parallel algorithm for iterating partitions of a finite set into subsets of a given cardinality
2020
Vestnik of Saint Petersburg University Applied Mathematics Computer Science Control Processes
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The first algorithm for each element of the original set determines the cardinality of the subset that it will fall into when partitioning. ...
-мат. наук, доц.; a.kovshov@spbu.ru A parallel algorithm for iterating partitions of a finite set into subsets of a given cardinality A. M. Kovshov St. ...
doi:10.21638/11702/spbu10.2020.105
fatcat:bshbnsfzb5cfbi2upjbrofqt6m
Page 3434 of Mathematical Reviews Vol. , Issue 87g
[page]
1987
Mathematical Reviews
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The expression a — (3); means that for every partition of the set [A]” of r-element subsets of a set A of cardinality a into 6 pieces, there is a set B of cardinality 6 with all its r-element subsets in ...
For example, if A is an uncountable cardinal, can we guarantee the existence of a homogeneous set of the same cardinality as A for every partition of the n-element subsets of A? ...
Some remarks on Kuratowski partitions
[article]
2017
arXiv
pre-print
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We introduce the notion of K-ideals associated with Kuratowski partitions and we prove that each κ-complete ideal on a measurable cardinal κ can be represented as a K-ideal. ...
We are very grateful for Reviewers and Editors who had the big influence of the final version of this paper.
Some remarks on Kuratowski partitions ...
Definitions and basic facts Let X be a topological space and let κ be a regular cardinal. Let F be a partition of X into meager sets. ...
arXiv:1706.08828v1
fatcat:36ks7yarufacxgt2swb6vg42ee
On special partitions of metric spaces
[article]
2020
arXiv
pre-print
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The main result of this paper is to show that, if κ is the smallest real-valued measurable cardinal not greater than 2^ℵ_0, then there exists a complete metric space of cardinality not greater than 2^κ ...
admitting a Kuratowski partition. ...
Let F be a K-partition of X of cardinality κ. Let I F be a K-ideal associated with F . By Fact 1, there exists a non-empty open set U ⊆ X such that I F ∩U is a precipitous ideal. ...
arXiv:2003.11017v4
fatcat:vkcnnhdkpbdp3aicvukhptzpdu
The b-domatic number of a graph
2013
Discussiones Mathematicae Graph Theory
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A partition P of G is a partition of its vertex set V . Its cardinality |P| is the number of classes of P. ...
A dominating set is divisible if it contains two disjoint dominating sets of G, indivisible otherwise. We denote by ω(G) the maximum cardinality of a clique of G. ...
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
The new operations on complete ideals
2019
Open Mathematics
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We introduce the notion of K-ideals associated with Kuratowski partitions. ...
Acknowledgement: The author is very grateful to the Reviewers for their comments which have raised the quality of this text and allowed to avoid many errors and inaccuracies. ...
The concept of Kuratowski partitions emerged when attempting to solve the problem set by K. ...
doi:10.1515/math-2019-0032
fatcat:gzwuurmkq5e47cmfjdhlkydvgq
Weak partition properties for infinite cardinals. I
1971
Proceedings of the American Mathematical Society
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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
Weak Partition Properties for Infinite Cardinals. I
1971
Proceedings of the American Mathematical Society
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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
Page 1527 of Mathematical Reviews Vol. 46, Issue 6
[page]
1973
Mathematical Reviews
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Say that a set is hereditarily of cardinality <y if (Vy eZ)
(card(y) <y). Denote by V, the class of all sets hereditarily
of cardinality <y. ...
Symbolic Logic 35 (1970), 410-428.
8839
The notation «—>(«)*, where « is a cardinal and « is an ordinal <x, means that whenever the set of increasing a-sequences of elements of « is partitioned into two ...
Connections between Kuratowski partitions of Baire spaces, measurable cardinals and precipitous ideals
[article]
2022
arXiv
pre-print
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In this paper we present a few properties of K-partitions, which are partitions of Baire spaces such that all subfamilies of such a partition sum to a set with the Baire property. ...
We also prove some connections between existence of K-partitions and existence of precipitous ideals as well as measurable cardinals. ...
The saturation of I -denoted by sat(I) -is the smallest cardinal such that all antichains in Y /I are of cardinality less than sat(I). Definition 4. Let I be an ideal on a set Y . ...
arXiv:2205.12653v1
fatcat:s7bqa7wkmfh3ng5zasb34fzv24
Bounds on the cardinality of partition
[article]
2018
arXiv
pre-print
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2009 pre-print arXiv:0711.2241v7 fatcat:jt2e5eyz5zhtdlckfdgqx4wuc4Other Versions
Let
Part A be the collection of partitions of a set
A . ...
INTRODUCTION
A partition of a set A is a collection of nonempty disjoint subsets of A whose union is Since this partition is unique, g is an injection. A . ...
arXiv:0711.2241v9
fatcat:lfngxxtl55cwdh5x4zblvj3rsi
Book Review: Combinatorial set theory
1979
Bulletin of the American Mathematical Society
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Globally, I very much like the spirit and the scope of the book. ...
Locally, more attention could have been paid to detail; there are many misprints, some mistatements of results, and some proofs need tightening. ...
Given cardinals K, À, and /x, and a positive integer «, the partition relation means that if A is a set of cardinality K and the set [A] n of (unordered) «-element subsets of A is partitioned into p pieces ...
doi:10.1090/s0273-0979-1979-14558-0
fatcat:5ram2bxhznenvbhcp7rniznjcm
Page 1274 of Mathematical Reviews Vol. , Issue 95c
[page]
1995
Mathematical Reviews
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In Section 2 they extend this relation to partitions of the set of all finite sequences in L,,.,,(A1 x --- x An), for an infinite sequence of cardinals /;, A2,---. ...
The polarized partition relation with these cardinals as parameters holds provided that for every partition of the Cartesian product A; x --- x A, into d parts, there is a sequence of sets H; € [A;]*', ...
A Comparison of Pair and Triple Partition Relations
[article]
2019
arXiv
pre-print
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This paper considers three different partition relations from partition calculus, two of which are pair relations and one of which is a triple relation. ...
An examination of the first partition relation and the ramification argument used to prove it will motivate questions regarding how to strengthen it. ...
Definition 8 (Partitions). For X a set and I an indexing set, a partition of X in I colors is a function χ : X → I. An r-partition of X in I colors is a partition of [X] r in I colors. ...
arXiv:1904.07790v1
fatcat:d2z5xcrc5nfxrf7a2mkgtbi6om
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