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

Super-simple (v,5,2)-designs

AbstractIn this paper we study the spectrum of super-simple (v,5,2)-designs. We show that a super-simple (v,5,2)-design exists if and only if v≡1 or 5(mod10),   v≠5,15, except possibly when v∈{75,95,115,135,195,215,231,285,365,385,515}

Ramsey numbers for sets of small graphs

AbstractThe Ramsey number r=r(G1-G2-⋯-Gm,H1-H2-⋯-Hn) denotes the smallest r such that every 2-coloring of the edges of the complete graph Kr contains a subgraph Gi with all edges of one color, of a subgraph Hi with all edges of a second color. These Ramsey numbers are determined for all sets of graphs with at most four vertices, and in the diagonal case (m=n,Gi=Hi) for all pairs of graphs, one with at most four and the other with five vertices, so as for all sets of graphs with five vertices