For a finite set X, we say that a set H ⊆ X crosses a partition P = (X1, . . . , X k ) of X if H intersects min(|H|, k) partition classes. If |H| ≥ k, this means that H meets all classes Xi, whilst for |H| ≤ k the elements of the crossing set H belong to mutually distinct classes. A set system H crosses P, if so does some H ∈ H. The minimum number of r-element subsets, such that every k-partition of an n-element set X is crossed by at least one of them, is denoted by f (n, k, r). The problem of

more »

... determining these minimum values for k = r was raised and studied by several authors, first by Sterboul in 1973 [Proc.