Title: Rado's Theorem
Abstract: Rado's Theorem is a theorem from combinatorics that
specializes to Halls' "Marriage Theorem" and also to the following
linear algebra statement: If $(Z_i)_{i \in I}$ is a family of finite
subsets of a vector space such that for each finite subset $J$ of $I$,
the elements in $\bigcup_{j \in J} Z_j$ span a subspace of dimension at
least $\#J$, then one can pick an element $v_i \in Z_i$ for each $i \in
I$ such that $(v_i)_{i \in I}$ is linearly independent. (I.e., the
obvious necessary condition is also sufficient.)
I will give the general statement and a proof and explain how it implies
the above-mentioned special cases.