Independent transversals for countable set systems

Abstract

Let F =〈 F i ¦i ∈ I 〉 be a system of subsets of S and let ℳ be an independence structure on S . A subsystem F K is ℳ- critical if it has an independent transversal versal and whenever B is a transversal of FK in ℳ, then B is a maximal independent subset of F(K) =U i ∈ K F i . It is shown that a necessary and sufficient condition for the existence of an independent transversal of a countable system F is that F i does not depend upon F(K) whenever F K is an ℳ-critical subsystem and i ∈ I\K .