Some Generalizations of the Problem of Distinct Representatives

Abstract

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 ai is in S i for (i = 1, 2, 3, … , n) and such that ai ≠ aj for i ≠ j , is that for k = 1, 2, 3, … , n, the union of any k of the sets S1, S 2, … , S n, contains at least k elements. The theorem has a number of consequences amongst which we list the following.