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Remark on a criterion for common transversals

Published online by Cambridge University Press:  18 May 2009

Hazel Perfect
Hazel Perfect
Affiliation:
Department of Pure Mathematics, The University, Sheffield
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All sets considered will be finite, and |x| will denote the cardinal number of the set X.

Let = (Ai:i∈I) be a family of subsets of a set E. A subset E′ ⊆ E is called a transversal of if there exists a bijection σ:E′→ I such that e ∈ Aσ(e) (e ∈ E′). According to a well-known theorem of P. Hall [2], the familyhas a transversal if and only iffor every subset I′ of I. Ford and Fulkerson [1] obtained (as a special case of a more general theorem) an analogous criterion for the existence of a common transversal (CT) of two families. We may state their result in the following terms.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 10 , Issue 1 , January 1969 , pp. 66 - 67
Copyright
Copyright © Glasgow Mathematical Journal Trust 1969

References

REFERENCES

1.Ford, L. R. Jr, and Fulkerson, D. R., Network flow and systems of representatives, Canad. J. Math. 10 (1958), 78–85.CrossRefGoogle Scholar
2.Hall, P., On representatives of subsets, J. London Math. Soc. 10 (1935), 26–30.CrossRefGoogle Scholar
3.Menger, K., Zur allgemeinem Kurventheorie, Fund. Math. 10 (1927), 96–115.CrossRefGoogle Scholar
4.Mirsky, L. and Perfect, H., Applications of the notion of independence to problems of combinatorial analysis, J. Combinatorial Theory 2 (1967), 327–57.CrossRefGoogle Scholar