Kirkman Packing and Covering Designs

Černý, Horák, and Wallis introduced a generalization of Kirkman’s Schoolgirl Problem to the case where the number of schoolgirls is not a multiple of three; they require all blocks to be of size three, except that each resolution class should contain either one block of size two (when \(v \equiv 2 \pmod{3}\)) or one block of size four (when \(v \equiv 1 \pmod{3}\)). We consider the problem of determining the maximum (resp. minimum) possible number of resolution classes such that any pair of elements (schoolgirls) is covered at most (resp. at least) once.