SATURATED R-UNIFORM HYPERGRAPHS

The following dual version of Turan's problem is considered: for a given r-uniform hypergraph F, determine the minimum number of edges in an r-uniform hypergraph H on n vertices, such that F not-a-subset-of H but a subhypergraph isomorphic to F occurs whenever a new edge (r-tuple) is added to H. For some types of F we find the exact value of the minimum or describe its asymptotic behavior as n tends to infinity; namely, for H(r)(r + 1, r), H(r)(2r - 2, 2) and H(r)(r + 1, 3), where H(r)(p, q) denotes the family of all r-uniform hypergraphs with p vertices and q edges. Several problems remain open.