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An application of a reduction method of R. Rado to the study of common transversals

Published online by Cambridge University Press:  26 February 2010

Colin McDiarmid
Colin McDiarmid
Affiliation:
Merton College, Oxford.
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Extract

A reduction method due to R. Rado [7] yields an elegant proof of the Halls' theorems on transversals of a family of sets (see for example [4]). We use here this method to give simple new proofs of some basic theorems on common transversals of a pair of families of sets.

Type
Research Article
Information
Mathematika , Volume 20 , Issue 1 , June 1973 , pp. 83 - 86
Copyright
Copyright © University College London 1973

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References

1. Brualdi, R. A., “A general theorem concerning common transversals”, Combinatorial Mathematics and its Applications Edited by Welsh, D. J. A. (1971).Google Scholar
2. Ford, L. R. and, Fulkerson, D. R., “Network flow and systems of representatives”, Canad. J. Math., 10 (1958), 78–84.CrossRefGoogle Scholar
3. Mirsky, L., “A theorem on common transversals”, Math. Annalen, 177 (1968), 49–53.CrossRefGoogle Scholar
4. Mirsky, L., Transversal Theory. (Academic Press, 1971).Google Scholar
5. McDiarmid, C. J. H., “Some problems in combinatorial theory” (to appear).Google Scholar
6. Pym, J. S., “The linking of sets in graphs”, J. London Math. Soc., 44 (1969), 542–550CrossRefGoogle Scholar
7. Rado, R., “Note on the transfinite case of Hall's theorem on representatives”, J. London Math. Soc 42 (1967), 321–324.CrossRefGoogle Scholar