Geodesic
Remark on a criterion for common transversals
Glasgow mathematical journal, Tome 10 (1969) no. 1, pp. 66-67

Voir la notice de l'article provenant de la source Cambridge University Press

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. 
Perfect, Hazel. Remark on a criterion for common transversals. Glasgow mathematical journal, Tome 10 (1969) no. 1, pp. 66-67. doi: 10.1017/S0017089500000550
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[1] 1.Ford, L. R. Jr, and Fulkerson, D. R., Network flow and systems of representatives, Canad. J. Math. 10 (1958), 78–85. Google Scholar | DOI

[2] 2.Hall, P., On representatives of subsets, J. London Math. Soc. 10 (1935), 26–30. Google Scholar | DOI

[3] 3.Menger, K., Zur allgemeinem Kurventheorie, Fund. Math. 10 (1927), 96–115. Google Scholar | DOI

[4] 4.Mirsky, L. and Perfect, H., Applications of the notion of independence to problems of combinatorial analysis, J. Combinatorial Theory 2 (1967), 327–57. Google Scholar | DOI

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