Minimum blocking sets for families of partitions

A 3-partition of an n-element set V is a triple of pairwise disjoint nonempty subsets X, Y, Z such that V=X Y Z. We determine the minimum size φ₃ (n) of a set E of triples such that for every 3-partition X, Y, Z of the set \1, , n\, there is some \x, y, z\ E with x X, y Y, and z Z. In particular, φ₃ (n) =n (n-2) {3}. For d>3, one may define an analogous number φd (n). We determine the order of magnitude of φd (n), and prove the following upper and lower bounds, for d>3: 2 n^d-1d! -o (n^d-1) φd (n) 0. 86 (d-1) !n^d-1+o (n^d-1).