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Keywords:
Boolean algebra; reaping number; partition
Boolean algebra; reaping number; partition
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 $\lceil \frac{\ell}{n} \rceil < \lceil \frac{k}{m-1} \rceil$.
References:
[1] Balcar B., Simon P.: On minimal $\pi$-character of points in extremally disconnected spaces. Topology Appl. 41 (1991), 133-145. MR 1129703
[2] Balcar B., Simon P.: Reaping number and $\pi$-character of Boolean algebras. preprint, 1991. MR 1189823 | Zbl 0766.06012
[3] Beslagić A., van Douwen E.K.: Spaces of nonuniform ultrafilters in spaces of uniform ultrafilters. Topology Appl. 35 (1990), 253-260. MR 1058805
[4] Dow A., Steprāns, Watson S.: Reaping numbers of Boolean algebras. preprint, 1992.
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