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On Existence of Distinct Representative Sets for Subsets of a Finite Set

Published online by Cambridge University Press:  20 November 2018

G. F. Clements
G. F. Clements*
Affiliation:
University of Colorado, Boulder, Colorado
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Let S be a finite set and let S1, S2, …, St be subsets of S, not necessarily distinct. Does there exist a set of distinct representatives (SDR) for S1, S2, …, St? That is, does there exist a subset {a1, a2, …, at} of S such that ai ∊ Si, 1 ≦ i ≦ t, and ai ≠ aj if i ≠ j? The following theorem of Hall [2; 6, p. 48] gives the answer.

THEOREM. The subsets S1, S2, …, St have an SDR if and only if for each s, 1 ≦ s ≦ t, |Si1 ∪ Si1 ∪ … ∪ Sis| ≧ s for each s-comhination {i1, i2, …, is} of the integers 1, 2, …, t.

(Here and below, |A| denotes the number of elements in A.)

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 22 , Issue 6 , 01 December 1970 , pp. 1284 - 1292
Copyright
Copyright © Canadian Mathematical Society 1970

References

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