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### Partitioning into Sets of Bounded Cardinality [chapter]

Mikko Koivisto
2009 Lecture Notes in Computer Science
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.  ...

### A parallel algorithm for iterating partitions of a finite set into subsets of a given cardinality

Alexander M. Kovshov, St. Petersburg State University
2020 Vestnik of Saint Petersburg University Applied Mathematics Computer Science Control Processes
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.  ...

### Page 3434 of Mathematical Reviews Vol. , Issue 87g [page]

1987 Mathematical Reviews
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]

Joanna Jureczko, Bogdan Węglorz
2017 arXiv   pre-print
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.  ...

### On special partitions of metric spaces [article]

Ryszard Frankiewicz, Joanna Jureczko
2020 arXiv   pre-print
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.  ...

### 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.  ...  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  ...

### The new operations on complete ideals

Joanna Jureczko
2019 Open Mathematics
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.  ...

### 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.  ...

### 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.  ...

### Page 1527 of Mathematical Reviews Vol. 46, Issue 6 [page]

1973 Mathematical Reviews
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]

Sławomir Kusiński
2022 arXiv   pre-print
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 .  ...

### Bounds on the cardinality of partition [article]

Kerry M. Soileau
2018 arXiv   pre-print
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 .  ...

### Book Review: Combinatorial set theory

James E. Baumgartner
1979 Bulletin of the American Mathematical Society
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  ...

### Page 1274 of Mathematical Reviews Vol. , Issue 95c [page]

1995 Mathematical Reviews
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]

Jonathon Eric Beers
2019 arXiv   pre-print
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.  ...
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