26 research outputs found
An exact Tur\'an result for tripartite 3-graphs
Mantel's theorem says that among all triangle-free graphs of a given order
the balanced complete bipartite graph is the unique graph of maximum size. We
prove an analogue of this result for 3-graphs. Let ,
and : for the
unique -free 3-graph of order and maximum size is the balanced
complete tripartite 3-graph (for it is
). This extends an old result of Bollob\'as
that is the unique 3-graph of maximum size with no copy of
or .Comment: 12 page
Pairwise Intersections and Forbidden Configurations
Let denote the maximum size of a family of subsets of an -element set for which there is no pair of subsets with , , , and . By symmetry we can assume and . We show that is if either or . We also show that is and is . This can be viewed as a result concerning forbidden configurations and is further evidence for a conjecture of Anstee and Sali. Our key tool is a strong stability version of the Complete Intersection Theorem of Ahlswede and Khachatrian, which is of independent interest
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