2 RRR 16 - 2009 1 On the size of a minimum transversal

Let n, d, and k be positive integers such that k ≥ 2, d ≥ 1, and n = kd. Furthermore, let N = {1, . . . , n} be the ground set and B(n, d) = {S ⊆ N : |S| = d} denote the family of all subsets of cardinality d in N ; obviously, |B(n, d)| = ( n d ) . A subfamily P (n, d) ⊆ B(n, d) that consists of k pairwise disjoint subsets is called a partition. Obviously, all sets of B(n, d) participate the same number of partitions. A subfamily T (n, d) ⊆ B(n, d) is called an (exact) transversal if it has a (unique) common set with each partition. Clearly, a transversal is exact if and only if every two of its sets intersect. From the above claims, it is not difficult to derive that |T (n, d)| ≥ ( n−1 d−1 ) for every transversal and the equality holds if and only if T (n, d) is exact. Thus, we obtain a new short proof of the famous Erdös-Ko-Rado Theorem in case n = kd, as well as the following application. Let F ⊆ B(n, d) be a set-family in which ` partitions are given explicitly, while the remaining sets of F are given by a membership oracle, to verify whether F contains another partition ( n−1 d−1 ) − ` tests might be needed.