Saturated r-uniform hypergraphs

AbstractThe following dual version of Turán'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 ⊄ 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 Hr(r + 1, r), Hr(2r −2, 2) and Hr(r + 1, 3), where Hr(p, q) denotes the family of all r-uniform hypergraphs with p vertices and q edges. Several problems remain open