Some approaches for solving the general (t,k)(t,k)-design existence problem and other related problems

In this short survey, some approaches that can be used to solve the generalized (t,k)(t,k)-design problem are considered. Special cases of the generalized (t,k)(t,k)-design problem include well-known conjectures for tt-designs, degree sequences of graphs and hypergraphs, and partial Steiner systems. Also described are some related problems such as the characterization of ff-vectors of pure simplicial complexes, which are well known but little understood. Some suggestions how enumerative and polyhedral techniques may help are also described.