Witness sets for families of binary vectors

Given a family ℛ ⊆ {0, 1}m of binary vectors of length m, as set W ⊆ {1,…,m} is called a witness set for r ∈ ℛ, if for all other r′ ∈ ℛ there exists a coordinate c ∈ W such that rc ≠ r′c. The smallest cardinality of a witness r ∈ ℛ is denoted w(r) = wℛ(r). In this note we show that ∑r∈ℛ w(r) = O(|ℛ|3/2) and constructions are given to show that this bound is tight. Further information is derived on the distribution of values of {w(r) | ∈ ℛ}.