f: � � [n]

Some special cases such as t-designs (g constant), partial Steiner systems (g constant and equal to 1), degree sequences of r-uniform hypergraphs (t = 1) and some related problems such as f -vectors of pure simplicial complexes [23, 26, 20] have received much attention during the last three decades. Yet we know very little about the problem. We do not even know whether or not it is NP-complete. Problem 1.1 is of great interest to statisticians since it includes the existence of designs as a special case. Statisticians are interested to know if there exists designs with some specific parameters. It would be of immense practical and theoretical value if one could find an algorithm (preferably polynomial-time) to solve Problem 1.1. In [36] a simple necessary and sufficient condition is given which solves Problem 1.1 if f takes values in Z. That paper also contains references to earlier work in this direction. In general, Problem 1.1 may not be solvable in polynomial time; in [10] it has been shown that some problems related to Problem 1.1 are NP-complete; but for many classes of designs it may be possible to obtain a polynomial-time algorithm. If we are allowed to repeat edges then the degree sequence problem is easily solvable in polynomial time and there are good characterizations [18, 34]. For graphs this problem is well-studied and there are many elegant characterizations. One of the well-known characterizations is due to Erdős-Gallai [13]. The book [28] gives 9 characterizations. For most of these characterizations a class of graphs called threshold graphs satisfy the characterizations in an extremal way. A graph is called threshold if it can be constructed from the one-vertex graph by repeatedly adding either an isolated vertex (i.e., a vertex non-adjacent to all previous vertices) or a dominating vertex (i.e., a vertex adjacent to all previous vertices). For graphs