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Sufficiency Conditions for the Existence of Transversals

Published online by Cambridge University Press:  20 November 2018

E. C. Milner
Affiliation:
University of Calgary, Calgary, Alberta
S. Shelah
Affiliation:
University of Jerusalem, Jerusalem, Israel
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A transversal of a family of non-empty sets is a 1-1 map

such that φ(v) ∊ Fv (vI) . A number of problems in combinatorial mathematics reduce to the question of whether or not a certain family of sets has a transversal. An up-to-date account of this theory is to be found in the book by Mirsky [9]. The best known result of this kind is the following theorem.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 1974

References

1. Alexandroff, and Urysohn, , Mémoire sur les espaces topologiques compacts, Verh. Nederl. Akad. Wentensch. Sect. I, 14, Nr. 1, SI (1929).Google Scholar
2. Bollobas, B. and Milner, E. C., A theorem in transversal theory, Bull. London Math. Soc. 5 (1973), 267270.CrossRefGoogle Scholar
3. Brualdi, R. A. and Scrimger, E. B., Exchange systems, matchings and transversals, J. Combinatorial Theory 5 (1968), 244257.CrossRefGoogle Scholar
4. Damerell, R. M. and Milner, E. C., Necessary and sufficient conditions for transversals of countable set systems (to appear in J. Combinatorial Theory).Google Scholar
5. Folkman, J., Transversals of infinite families with finitely many infinite members, RAND Corporation Memorandum RM-5676-PR, 1968; J. Combinatorial Theory 9 (1970), 200220.Google Scholar
6. Hall, Marshall, Jr., Distinct representatives of subsets, Bull. Amer. Math. Soc. 54 (1948), 922926.CrossRefGoogle Scholar
7. Hall, P., On representatives of subsets, J. London Math. Soc. 10 (1935), 2630.CrossRefGoogle Scholar
8. König, J., Graphok es matrixok, Mat. Lapok 38 (1931), 116119.Google Scholar
9. Mirsky, L., Transversal theory (Academic Press, New York, 1971).Google Scholar
10. Williams, C. St. J. A. Nash, Proceedings of conference in combinatorics and graph theory, Oxford, 1972.Google Scholar
11. Neumer, W., Verallgemeinerung eines Satzes von Alexandrqff and Urysohn, Math. Z. 54 (1951), 254261.CrossRefGoogle Scholar
12. Rado, R., Note on the transfinite case of Hall's theorem on representatives, J. London Math. Soc. 4% (1967), 321-324. 13 S. Shelah, A substitute for HalVs theorem for families with infinite sets, J. Combinatorial Theory (A) 16 (1974), 199208.Google Scholar
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