Two conjectures of Demetrovics, Füredi, and Katona, concerning partitions

AbstractIt is possible to find n partitions of an n-element set whose pairwise intersections are just all atoms of the partition lattice? Demetrovics, Füredi and Katona [4] verified this for all n ≡ 1 or 4 (mod 12) by constructing a series of special Mendelsohn Triple Systems. They conjectured that such triple systems exist for all n ≡ 1 (mod 3) and that the problem on the partitions has a solution for all n ⩾ 7. We prove that both conjectures are ture, except for finitely many n