Let $A$ be an abelian group. Recall that $A$ is

($\bullet$) a *Whitehead group* if $\text{Ext}(A,\mathbb Z)=0$,

($\bullet$) a *free abelian group* if $\text{Ext}(A,D)=0$ for every abelian group $D$.

Now fix a cardinal $\mathfrak k$ and consider the following property of $A$:

($\star$) $\text{Ext}(A,D)=0$ **for every torsion-free abelian group $D$ with cardinality at most $\mathfrak k$**.

What are the groups $A$ satisfying condition ($\star$)?

Observe: If the cardinality of $A$ itself is at most $\mathfrak k$ then ($\star$) implies that $A$ is free abelian. (In such case $A$ has a free resolution $0\rightarrow D\rightarrow F\rightarrow A\rightarrow 0$ with $D$ as in ($\star$) and this sequence splits by the assumptions.) What happens if the cardinality of $A$ is greater than $\mathfrak k$?