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 $\lceil \frac{\ell}{n} \rceil < \lceil \frac{k}{m-1} \rceil$.
@article{118593,
author = {Claude Laflamme},
title = {Partitions of $k$-branching trees and the reaping number of Boolean algebras},
journal = {Commentationes Mathematicae Universitatis Carolinae},
volume = {34},
year = {1993},
pages = {397-399},
zbl = {0783.06009},
mrnumber = {1241749},
language = {en},
url = {http://dml.mathdoc.fr/item/118593}
}
Laflamme, Claude. Partitions of $k$-branching trees and the reaping number of Boolean algebras. Commentationes Mathematicae Universitatis Carolinae, Tome 34 (1993) pp. 397-399. http://gdmltest.u-ga.fr/item/118593/
On minimal $\pi$-character of points in extremally disconnected spaces, Topology Appl. 41 (1991), 133-145. (1991) | MR 1129703
Reaping number and $\pi$-character of Boolean algebras, preprint, 1991. | MR 1189823 | Zbl 0766.06012
Spaces of nonuniform ultrafilters in spaces of uniform ultrafilters, Topology Appl. 35 (1990), 253-260. (1990) | MR 1058805
Reaping numbers of Boolean algebras, preprint, 1992.