I found the following unproved statements in Shelah and Mekler's article "On the consistency strength of "every stationary set reflects"".
Let $\kappa$ be the least cardinal such that there exists a $\kappa$-free abelian group which is not $\kappa^+$-free. By a well known argument, $\kappa$ is either the successor of a singular cardinal or an inaccessible cardinal.
It is easy to see (and well known) that if every stationary set reflects in a regular cardinal then every $\kappa$-free abelian group is $\kappa^+$-free.
Can someone please give references for the above statements? In particulary I am deeply interested in the first one, but I couldn't find any introductory material. The wonderful book "Almost free modules" by Paul Eklof strangely doesn't mention the result (or I couldn't find it!).
Thank you in advance.
The paper
that you mention in the question actually provides the relevant references and explains the key idea of the argument. Note first that $\kappa$ is assumed regular.
They refer to
In that paper, Eklof essentially shows that if $\kappa$ is regular and there is a stationary subset of $\kappa$ that does not reflect, then there is a $\kappa$-free not $\kappa^+$-free abelian group. This means that if we want the opposite, every stationary subset of $\kappa$ must reflect.
The point then is to note that $S^{\tau^+}_\tau$ is non-reflecting whenever $\tau$ is regular. Here $S^{\tau^+}_\tau$ is the set of ordinals smaller than $\tau^+$ of cofinality $\tau$. To see this, let $\alpha\in S^{\tau^+}_\tau$, fix a club subset $C$ of $\alpha$ of type $\tau$, and note that $C'$ cannot meet $S^{\tau^+}_\tau$ since no point of $C'$ has cofinality $\tau$.
This leaves us with $\kappa$ being inaccessible or the successor of a singular, since all other regular cardinals are of the form $\kappa^+$ for $\kappa$ regular.