11.90 (1990)

Let $\mathfrak{V}$ be a variety of groups and let $k$ be an infinite cardinal number. A group $G \in \mathfrak{V}$ of rank $k$ is called almost free in $\mathfrak{V}$ if each of its subgroups of rank less than $k$ is contained in a subgroup of $G$ which is free in $\mathfrak{V}$. For what $k$ do there exist almost free but not free in $\mathfrak{V}$ groups of rank $k$?