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 ,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 nᵈ⁻¹d! -o(nᵈ⁻¹) ≤ φd(n) ≤ 0.86/(d-1)!nᵈ⁻¹+o(nᵈ⁻¹).