Partitions of k -branching trees and the reaping number of Boolean algebras

Claude Laflamme

Commentationes Mathematicae Universitatis Carolinae (1993)

  • Volume: 34, Issue: 2, page 397-399
  • ISSN: 0010-2628

Abstract

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The reaping number 𝔯 m , n ( 𝔹 ) of a Boolean algebra 𝔹 is defined as the minimum size of a subset 𝒜 𝔹 { 𝐎 } such that for each m -partition 𝒫 of unity, some member of 𝒜 meets less than n elements of 𝒫 . We show that for each 𝔹 , 𝔯 m , n ( 𝔹 ) = 𝔯 m n - 1 , 2 ( 𝔹 ) as conjectured by Dow, Steprāns and Watson. The proof relies on a partition theorem for finite trees; namely that every k -branching tree whose maximal nodes are coloured with colours contains an m -branching subtree using at most n colours if and only if n < k m - 1 .

How to cite

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Laflamme, Claude. "Partitions of $k$-branching trees and the reaping number of Boolean algebras." Commentationes Mathematicae Universitatis Carolinae 34.2 (1993): 397-399. <http://eudml.org/doc/21949>.

@article{Laflamme1993,
abstract = {The reaping number $\mathfrak \{r\}_\{m,n\}(\{\mathbb \{B\}\})$ of a Boolean algebra $\{\mathbb \{B\}\}$ is defined as the minimum size of a subset $\{\mathcal \{A\}\} \subseteq \{\mathbb \{B\}\}\setminus \lbrace \{\mathbf \{O\}\}\rbrace $ such that for each $m$-partition $\mathcal \{P\}$ of unity, some member of $\mathcal \{A\}$ meets less than $n$ elements of $\mathcal \{P\}$. We show that for each $\{\mathbb \{B\}\}$, $\mathfrak \{r\}_\{m,n\}(\mathbb \{B\}) = \mathfrak \{r\}_\{\lceil \frac\{m\}\{n-1\} \rceil ,2\}(\mathbb \{B\})$ as conjectured by Dow, Steprāns and Watson. The proof relies on a partition theorem for finite trees; namely that every $k$-branching tree whose maximal nodes are coloured with $\ell $ colours contains an $m$-branching subtree using at most $n$ colours if and only if $\lceil \frac\{\ell \}\{n\} \rceil < \lceil \frac\{k\}\{m-1\} \rceil $.},
author = {Laflamme, Claude},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {Boolean algebra; reaping number; partition; -partition of unity; reaping number; -branching tree},
language = {eng},
number = {2},
pages = {397-399},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Partitions of $k$-branching trees and the reaping number of Boolean algebras},
url = {http://eudml.org/doc/21949},
volume = {34},
year = {1993},
}

TY - JOUR
AU - Laflamme, Claude
TI - Partitions of $k$-branching trees and the reaping number of Boolean algebras
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1993
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 34
IS - 2
SP - 397
EP - 399
AB - The reaping number $\mathfrak {r}_{m,n}({\mathbb {B}})$ of a Boolean algebra ${\mathbb {B}}$ is defined as the minimum size of a subset ${\mathcal {A}} \subseteq {\mathbb {B}}\setminus \lbrace {\mathbf {O}}\rbrace $ such that for each $m$-partition $\mathcal {P}$ of unity, some member of $\mathcal {A}$ meets less than $n$ elements of $\mathcal {P}$. We show that for each ${\mathbb {B}}$, $\mathfrak {r}_{m,n}(\mathbb {B}) = \mathfrak {r}_{\lceil \frac{m}{n-1} \rceil ,2}(\mathbb {B})$ as conjectured by Dow, Steprāns and Watson. The proof relies on a partition theorem for finite trees; namely that every $k$-branching tree whose maximal nodes are coloured with $\ell $ colours contains an $m$-branching subtree using at most $n$ colours if and only if $\lceil \frac{\ell }{n} \rceil < \lceil \frac{k}{m-1} \rceil $.
LA - eng
KW - Boolean algebra; reaping number; partition; -partition of unity; reaping number; -branching tree
UR - http://eudml.org/doc/21949
ER -

References

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  1. Balcar B., Simon P., On minimal π -character of points in extremally disconnected spaces, Topology Appl. 41 (1991), 133-145. (1991) Zbl0752.54013MR1129703
  2. Balcar B., Simon P., Reaping number and π -character of Boolean algebras, preprint, 1991. Zbl0766.06012MR1189823
  3. Beslagić A., van Douwen E.K., Spaces of nonuniform ultrafilters in spaces of uniform ultrafilters, Topology Appl. 35 (1990), 253-260. (1990) Zbl0716.54002MR1058805
  4. Dow A., Steprāns, Watson S., Reaping numbers of Boolean algebras, preprint, 1992. 

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