@MISC{HALLMAR1.MIZ,
  AUTHOR = {Romanowicz, Ewa and Grabowski, Adam},
  TITLE = {The {H}all {M}arriage {T}heorem},
  SECTION1 = {Preliminaries},
  SECTION2 = {Union of Finite Sequences},
  SECTION3 = {Cut Operation for Finite Sequences},
  SECTION4 = {System of Different Representatives and Hall Property},
  SECTION5 = {Reductions and Singlifications of Finite Sequences},
  SECTION6 = {Rado's Proof of the Hall Marriage Theorem},
  DAY = {11},
  MONTH = {May},
  YEAR = {2004},
  ADDRESS1 = {University of Bia{\l}ystok},
  ADDRESS2 = {University of Bia{\l}ystok},
  SUMMARY = {The Marriage Theorem, as credited to Philip Hall \cite{Hall:1935},
  gives
  the necessary and sufficient condition allowing us to
  select a distinct element from each of a finite collection
  $\{A_i\}$  of $n$ finite subsets. This selection, called
  a set of different representatives (SDR), exists if and only if
  the marriage condition (or Hall condition) is satisfied:
  $$\forall_{J\subseteq\{1,\dots,n\}}|\bigcup_{i\in J} A_i|\geq |J|.$$
  The proof which is given in this article (according to Richard Rado,
  1967) is based on the lemma that for finite sequences
  with non-trivial elements which satisfy Hall property there exists
  a reduction (see Def. 5) such that Hall property again holds
  (see Th.~29 for details).}}

