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An Extension of Hall's Theorem
I. Pinelis
Department of Mathematical Sciences, Michigan Technological University, Houghton, MI 49931, USA
ipinelis@mtu.edu
Annals of Combinatorics 6 (1) p.103-106 March, 2002
AMS Subject Classification: 05A05, 03E05, 04A20, 05A18, 05C50
Abstract:
Let and be two (possibly infinite) families of finite sets. Let denote the closure of the set of the pairs with respect to the component-wise union and intersection operations. Then there exists an injective map such that for every i if, and only if, for every pair .
Keywords: Hall's theorem, injectivity, systems of distinct representatives, infinite families of sets

References:

1. R. Aharoni and P. Haxell, HallĄ¯s theorem for hypergraphs, J. Graph Theory 35 (2000) 83¨C88.

2. J.M. Hall, Combinatorial Theory, Second Edition, Wiley, New York, 1986.

3. I. Pinelis, Multilinear generalizations of the direct and reverse Stolarsky inequalities, preprint.