If S 1 , S 2 , S 3 , … , S n are subsets of a set M then it is known that a necessary and sufficient condition that it is possible to choose representatives at such that a i is in S i for ( i = 1, 2, 3, … , n ) and such that a i ≠ a j for i ≠ j , is that for k = 1, 2, 3, … , n , the union of any k of the sets S 1 , S 2 , … , S n , contains at least k elements. The theorem has a number of consequences amongst which we list the following.
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Mendelsohn et al. (1958) studied this question.