| Title:
|
Partitions of $k$-branching trees and the reaping number of Boolean algebras (English) |
| Author:
|
Laflamme, Claude |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
34 |
| Issue:
|
2 |
| Year:
|
1993 |
| Pages:
|
397-399 |
| . |
| Category:
|
math |
| . |
| Summary:
|
The reaping number $\frak r_{m,n}({\Bbb B})$ of a Boolean algebra ${\Bbb B}$ is defined as the minimum size of a subset ${\Cal A} \subseteq {\Bbb B}\setminus \{{\bold O}\}$ such that for each $m$-partition $\Cal P$ of unity, some member of $\Cal A$ meets less than $n$ elements of $\Cal P$. We show that for each ${\Bbb B}$, $\frak r_{m,n}(\Bbb B) = \frak r_{\lceil \frac{m}{n-1} \rceil,2}(\Bbb B)$ 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 $\lceil \frac{\ell}{n} \rceil < \lceil \frac{k}{m-1} \rceil$. (English) |
| Keyword:
|
Boolean algebra |
| Keyword:
|
reaping number |
| Keyword:
|
partition |
| MSC:
|
05C05 |
| MSC:
|
05C15 |
| MSC:
|
05C90 |
| MSC:
|
06E05 |
| MSC:
|
06E10 |
| idZBL:
|
Zbl 0783.06009 |
| idMR:
|
MR1241749 |
| . |
| Date available:
|
2009-01-08T18:04:28Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/118593 |
| . |
| Reference:
|
[1] Balcar B., Simon P.: On minimal $\pi$-character of points in extremally disconnected spaces.Topology Appl. 41 (1991), 133-145. MR 1129703 |
| Reference:
|
[2] Balcar B., Simon P.: Reaping number and $\pi$-character of Boolean algebras.preprint, 1991. Zbl 0766.06012, MR 1189823 |
| Reference:
|
[3] Beslagić A., van Douwen E.K.: Spaces of nonuniform ultrafilters in spaces of uniform ultrafilters.Topology Appl. 35 (1990), 253-260. MR 1058805 |
| Reference:
|
[4] Dow A., Steprāns, Watson S.: Reaping numbers of Boolean algebras.preprint, 1992. |
| . |
