On Kelley’s Intersection Numbers

We introduce a notion of weak intersection number of a collection of sets, modifying the notion of intersection number due to J.L. Kelley, and obtain an analogue of Kelley’s characterization of Boolean algebras which support a finitely additive strictly positive measure. We also consider graph-theoretic reformulations of the notions of intersection number and weak intersection number. Kelley [4] gave a necessary and sufficient condition for a Boolean algebra B to have a finitely additive strictly positive measure. A finitely additive nonnegative measure μ on B is strictly positive if, for B ∈ B, μ(B) = 0 if and only if B is the zero element of B. Kelley’s condition involves the notion of intersection number of a collection of elements of B, or equivalently, of a collection of sets. In the present note we introduce a related notion of weak intersection number , obtain an analogue of Kelley’s condition in Theorem 2, and investigate the relationship between these two kinds of intersection numbers (Theorems 2 and 5). We will also reformulate these notions in graph-theoretic terms. In particular, in Theorem 6 we shall establish a connection between the notions of (Kelley’s) intersection number and fractional chromatic number for hypergraphs. Let A = 〈A1, A2, . . . , An〉, where n > 0, be a sequence of elements of B and let I ⊆ {1, 2, . . . , n} be of maximum cardinality such that ⋂