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 $\mathfrak k$?