On k-transversals

AbstractA k-transversal of a family of sets H is a function g: H → [∪ H]k + 1 satisfying |∪ g[F]| ⩾ |F| + k for every nonempty subfamily F of H and g(H) ⊆ H for every H ϵ H. It is proved that a (possibly infinite) family H has a k-transversal if and only if the family (H⧹S: H ϵ H) has a transversal for every S ϵ [∪ H]k. One corollary is that a family of infinite sets has a k-transversal if and only if it has a transversal. It is also shown that if l ⩽ k and f(H) ϵ [H]l for each H ϵ H and H has a k-transversal, then it has a k-transversal g such that g(H) ⊇ f(H) for every H ϵ H