The linear refinement number and selection theory

The linear refinement number lr is the minimal cardinality of a centered family in [ω]^ω such that no linearly ordered set in ([ω]^ω,⊆^∗) refines this family. The linear excluded middle number lr is a variation of lr. We show that these numbers estimate the critical cardinalities of a number of selective covering properties. We compare these numbers to the classical combinatorial cardinal characteristics of the continuum. We prove that lr = lŗ = d in all models where the continuum is at most ℵ₂, and that the cofinality of lr is uncountable. Using the method of forcing, we show that lr and lŗ are not provably equal to d, and rule out several potential bounds on these numbers. Our results solve a number of open problems.