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

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
.

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.