Reaping Numbers of Boolean Algebras

A subset A 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 ≠ 0}| ⩾ m for all a ∈ A . The reaping number r n , m (B) of a Boolean algebra B is the minimum cardinality of a set A ⊂ B∖{0} which cannot be ( n , m )‐reaped. It is shown that for each n ∈ω, there is a Boolean algebra B such that r n +1,2 (B) ≠ r n ,2 (B). Also, { r n , m (B): m ⩽ n ∈ ω} consists of at most two consecutive cardinals. The existence of a Boolean algebra B such that r n , m (B) ≠ r n ′, m ′ (B) is equivalent to a statement in finite combinatorics which is also discussed.