2-cancellative hypergraphs and codes

A family of sets F (and the corresponding family of 0-1 vectors) is called t-cancellative if for all distict t+2 members A_1,... A_t and B,C from F the union of A_1,..., A_t and B differs from the union of A_1, ..., A_t and C. Let c(n,t) be the size of the largest t-cancellative family on n elements, and let c_k(n,t) denote the largest k-uniform family. We significantly improve the previous upper bounds, e.g., we show c(n,2) n_0). Using an algebraic construction we show that the order of magnitude of c_{2k}(n,2) is n^k for each k (when n goes to infinity).Comment: 20 page

A Better Bound for Locally Thin Set Families

AbstractA family of subsets of an n-set is 4-locally thin if for every quadruple of its members the ground set has at least one element contained in exactly 1 of them. We show that such a family has at most 20.4561n members. This improves on our previous results with Noga Alon. The new proof is based on a more careful analysis of the self-similarity of the graph associated with such set families by the graph entropy bounding technique

On colorful edge triples in edge-colored complete graphs

An edge-coloring of the complete graph Kn we call F-caring if it leaves no F-subgraph of Kn monochromatic and at the same time every subset of |V(F)| vertices contains in it at least one completely multicolored version of F. For the first two meaningful cases, when F=K1,3 and F=P4 we determine for infinitely many n the minimum number of colors needed for an F-caring edge-coloring of Kn. An explicit family of 2⌈log2n⌉ 3-edge-colorings of Kn so that every quadruple of its vertices contains a totally multicolored P4 in at least one of them is also presented. Investigating related Ramsey-type problems we also show that the Shannon (OR-)capacity of the Grötzsch graph is strictly larger than that of the five length cycle