Abstract
A system of distinct representatives (SDR) of a family F = (A1, . . . , An) is a sequence (a1, . . . , an) of n distinct elements with ai ∈ Ai for 1 ⩾ i ⩾ n. Denote by N(F) the number of SDR of a family F, two SDR are considered distinct if they are different in at least one component. F = (A1, . . . , An) is a (t, n)-family if | ∪i∈IAi| > ⩽ |I| + t for any non-empty subset I ⊆ {1, . . . , n} . A theorem of P. Hall says that N(F) ⩽ 1 iff F is a (0, n)-family. Let M(t, n) = min {N(F): F is a (t, n)-family}. In this paper we prove that M(l, n) = n + 1 and M(2, n) = n2 + n + 1. We also determine all (t, n)-families F with N(F) = M(t, n) for t = 0, l, 2.
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