Let 0 be any fixed 3-uniform hypergraph. For a 3-uniform hypergraph we define 0( ) to be the maximum size of a set of pairwise triple-disjoint copies of 0 in . We say a function from the set of copies of 0 in to [0, 1] is a fractional 0-packing of if ¥ e ( ) 1 for every triple e of . Then * 0( ) is defined to be the maximum value of ¥ 0 over all fractional 0-packings of . We show that * 0( ) 0(...

A k-uniform hypergraph with vertex set V and edge set E is called t-subset-regular if every t-element subset of V lies in the same number of elements of E. In this paper we establish a necessary condition on n for there to exist a t-subset-regular selfcomplementary k-uniform hypergraph with n vertices. In addition, we show that this necessary condition is also sufficient in the case k = 3 and t...

We give lower bounds on the maximum possible girth of an r-uniform, d-regular hypergraph with at most n vertices, using the definition of a hypergraph cycle due to Berge. These differ from the trivial upper bound by an absolute constant factor (viz., by a factor of between 3/2 + o(1) and 2 + o(1)). We also define a random r-uniform ‘Cayley’ hypergraph on the symmetric group Sn which has girth Ω...

Let H’ be an r-uniform hypergraph. Let g=g(n; F) be the minimal integer so that any r-uniform hypergraph on n vertices and more than g edges contains a subgraph isomorphic to H’. Let e=f(n; H’, en) denote the minimal integer such that every r-uniform hypergraph on n vertices with more than e edges and with no independent set of .sn vertices contains a subgraph isomorphic to H’. We show that if ...

In this notewe give a new effective proofmethod for the equivalence of the notions of testability and nondeterministic testability for uniform hypergraph parameters. We provide the first effective upper bound on the sample complexity of any nondeterministically testable r-uniform hypergraph parameter as a function of the sample complexity of its witness parameter for arbitrary r. The dependence...

We say that a hypergraph H is hamiltonian path (cycle) saturated if H does not contain an open (closed) hamiltonian chain but by adding any new edge we create an open (closed) hamiltonian chain in H. In this paper we ask about the smallest size of an r-uniform hamiltonian path (cycle) saturated hypergraph, mainly for r = 3. We present a construction of a family of 3-uniform path (cycle) saturat...

Hypergraph entropy is an information theoretic functional on a hypergraph with a probability distribution on its vertex set. It is sub-additive with respect to the union of hypergraphs. In case of simple graphs, exact additivity for the entropy of a graph and its complement with respect to every probability distribution on the vertex set gives a characterization of perfect graphs. Here we inves...

The r-expansion G of a graph G is the r-uniform hypergraph obtained from G by enlarging each edge of G with a vertex subset of size r − 2 disjoint from V (G) such that distinct edges are enlarged by disjoint subsets. Let exr(n, F ) denote the maximum number of edges in an r-uniform hypergraph with n vertices not containing any copy of the r-uniform hypergraph F . Many problems in extremal set t...

Let F be a strictly k-balanced k-uniform hypergraph with e(F ) ≥ |F | − k + 1 and maximum co-degree at least two. The random greedy F -free process constructs a maximal F -free hypergraph as follows. Consider a random ordering of the hyperedges of the complete k-uniform hypergraph K n on n vertices. Start with the empty hypergraph on n vertices. Successively consider the hyperedges e of K n in ...