Still another exercise on Reinhard Diestel Graph Theory, GTM 173, edition 3 (on page 51)
Let $A$ be a finite set with subsets $A_1, \cdots, A_n$, and let $d_1, \cdots, d_n \in \mathbb N$. Show that there are disjoint subsets $D_k \subseteq A_k$, with $|D_k| =d_k$ for all $k \leq n$, if and only if $|\cup_{i\in I}A_i| \geq \sum_{i \in I} d_i$ for all $I \subseteq \{ 1,\cdots,n\}$.
This "if and only if" statement is clear in one direction since for the properly chosen $D_k$, $|\cup_{i\in I}A_i| \geq |\cup_{i\in I}D_i| = \sum_{i \in I} d_i$. But the proof of the other direction seems more complicated to me. I tried to prove it by induction, but failed.
Since I am a beginner of graph theory, I just take it as an exercise of set theory, and cannot think it in the way of graphs. Maybe graph theoretic methods are favorable. Longing for your advice. Thanks very much.