For each infinite cardinal number κ, let Ω(κ) be the supremum of the cardinalities of chains of subsets of a set of cardinality κ. (Ω(κ) is equal to what has been called ded(κ) in the literature.) Let K be a field and V a vector space over K. Let Λ(V ) be the supremum of the cardinalities of chains of vector subspaces of V . Let the dimension of V as a vector space over K be the infinite cardinal number κ. Then Ω(κ) ≤ Λ(V ) ≤ Ω(|V |), and so Λ(V ) > κ, contrary to a result of Menth. If, in addition, K is either finite or infinite with |K| ≤ κ, then Ω(κ) = Ω(|V |) (= Λ(V )).