Reaping Numbers of Boolean Algebras

A subsetA of a Boolean algebra B is said to be (n,m)reaped if there is a partition of unity P ⊂ B of size n such that |{b ∈ P : b∧a 6= ∅}| ≥ m for all a ∈ A. The reaping number rn,m(B) of a Boolean algebra B is the minimum cardinality of a set A ⊂ B\ {0} such which cannot be (n,m)-reaped. It is shown that, for each n ∈ ω, there is a Boolean algebra B such that rn+1,2(B) 6= rn,2(B). Also, {rn,m(B) : {n,m} ⊆ ω} consists of at most two consecutive integers. The existence of a Boolean algebra B such that rn,m(B) 6= rn′,m′(B) is equivalent to a statement in finite combinatorics which is also discussed.