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

Claude Laflamme (1993)

Commentationes Mathematicae Universitatis Carolinae

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 ℓ colours contains an m -branching subtree using at most n colours if and only if ⌈ ℓ n ⌉ < ⌈ k m - 1 ⌉ .

Perfect compactifications of frames

Dharmanand Baboolal (2011)

Czechoslovak Mathematical Journal

Perfect compactifications of frames are introduced. It is shown that the Stone-Čech compactification is an example of such a compactification. We also introduce rim-compact frames and for such frames we define its Freudenthal compactification, another example of a perfect compactification. The remainder of a rim-compact frame in its Freudenthal compactification is shown to be zero-dimensional. It is shown that with the assumption of the Boolean Ultrafilter Theorem the Freudenthal compactification...

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