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Disjoint Transversals of Subsets

Published online by Cambridge University Press:  20 November 2018

P. J. Higgins
P. J. Higgins*
Affiliation:
Harvard University
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Let A1, A2,… , An be a finite collection of subsets (not necessarily distinct) of a set A. By a transversal of A1, A2,… , An we shall mean a set of n distinct elements a1, a2,… , an of A such that, for some permutation i1 i2, … , in of the integers 1, 2, … , n,

More generally, we shall say that the set {a1, a2, … , ar}, (r ≤ n) is a partial transversal oi A1, A2, … An of length r if (i) a1, a2, … , ar are distinct elements of A and (ii) there exists a set of distinct integers i1, i2… , irsuch that

A well-known theorem of P. Hall (2) states that the sets A1, A2… , An have a transversal (of length n) if, and only if, every k of them contain collectively at least k distinct elements (k = 1, 2, … , n).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 11 , 1959 , pp. 280 - 285
Copyright
Copyright © Canadian Mathematical Society 1959

References

1. Gale, David, A theorem on flows in networks, Pacific J. Math., 7 (1957), 1073-82.Google Scholar
2. Hall, Philip, On representatives of subsets, J. Lond. Math. Soc, 10 (1935), 26–30.Google Scholar
3. Ore, Oystein, Graphs and matching theorems, Duke Math. J., 22 (1955), 625-39.Google Scholar
4. Ryser, H. J., Combinatorial properties of matrices of zeros and ones, Can. J. Math., 9 (1957), 371–7.Google Scholar