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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Hall's theorem revisited


Author: Zhi-Wei Sun
Journal: Proc. Amer. Math. Soc. 129 (2001), 3129-3131
MSC (2000): Primary 05A05; Secondary 05C20
Published electronically: May 6, 2001
MathSciNet review: 1840120
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Abstract: Let $A_{1},\cdots ,A_{n} (n>1)$ be sets. By a simple graph-theoretic argument we show that any set of distinct representatives of $\{A_{i}\}_{i=1}^{n-1}$can be extended to a set of distinct representatives of $\{A_{i}\}_{i=1}^{n}$in more than $\min _{n\in I\subseteq \{1,\cdots ,n\}} (\vert\bigcup _{i\in I}A_{i}\vert-\vert I\vert)$ ways. This yields a natural induction proof of the well-known theorem of P. Hall.


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Additional Information

Zhi-Wei Sun
Affiliation: Department of Mathematics, Nanjing University, Nanjing 210093, People’s Republic of China
Email: zwsun@nju.edu.cn

DOI: http://dx.doi.org/10.1090/S0002-9939-01-06215-3
PII: S 0002-9939(01)06215-3
Received by editor(s): March 31, 2000
Received by editor(s) in revised form: October 30, 2000
Published electronically: May 6, 2001
Additional Notes: This research was supported by the Teaching and Research Award Fund for Outstanding Young Teachers in Higher Education Institutions of MOE, and the National Natural Science Foundation of P. R. China.
Communicated by: John R. Stembridge
Article copyright: © Copyright 2001 American Mathematical Society