= Solution
Form a <bipartite graph> with left labels $1,\ldots,m$, right set $E$, and an <edge> $i\!\mathbin{-}\!e$ exactly when $e\in S_i$. A <matching> covering all left labels is precisely a choice of distinct representatives. Its neighborhood condition is
$$
\boxed{\left|\bigcup_{i\in I}S_i\right|\ge|I|\quad\text{for every }I\subseteq\{1,\ldots,m\}.}
$$
Thus this is necessary and sufficient. The underlying set $E$ may be infinite; only the number of sets to be represented is finite.
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