We consider the problem of finding a uniform hypergraph that satisfies cut demands defined by a symmetric crossing supermodular set function. We give min-max formulas for both the degree specified and the minimum cardinality problem. These results include as a special case a formula on the minimum number of r-hyperedges whose addition to an initial hypergraph will make it k-edge-connected.

We study the minimum number of complete r-partite r-uniform hypergraphs needed to partition the edges of the complete r-uniform hypergraph on n vertices and we improve previous results of Alon.

Denote by an l-component a connected b-uniform hypergraph with k edges and k(b− 1)− l vertices. We prove that the expected number of creations of l-component during a random hypergraph process tends to 1 as l and b tend to ∞ with the total number of vertices n such that l = o (

A 3-uniform hypergraph is called a minimum 3-tree if for any 3-coloring of its vertex set there is a heterochromatic edge and the hypergraph has the minimum possible numberof edges. Here we s h o w that the numberof edges in such 3-tree is l n(n;2) 3 m for any n umberof vertices n 3 4 (mod 6):

Given a simple graph Γ, we describe a “lifting” to a 3-uniform hypergraph φ(Γ) that sends the complement of Γ to the complement of φ(Γ). We consider the effects of this lifting on cycles, complete subhypergraphs, and complete subhypergraphs missing a single hyperedge. Our results lead to natural lower bounds for some hypergraph Ramsey numbers.

A path-hypergraph P (3)(2, 4) is the uniform hypergraph of rank 3, having four vertices and two edges, with exactly two vertices in common. In this paper, the intersection problem among P (3)(2, 4)-designs of order v is studied and completely solved for every admissible v. Mathematics Subject Classification: 05B05

A two-coloring of the vertices X of the hypergraph H = ( X , o °) by red and blue has discrepancy d if d is the largest difference between the number of red and blue points in any edge. A two-coloring is an equipartition of H if it has discrepancy 0, i.e., every edge is exactly half red and half blue. Letf(n) be the fewest number of edges in an n-uniform hypergraph (all edges have size n) havin...