f-vectors of pure complexes and pure multicomplexes of rank three

Necessary and sufficient conditions are established for an integer vector to be the f -vector of some pure simplicial complex of rank three, and also for an integer vector to be the f -vector of some pure simplicial multicomplex of rank three. For specified numbers of sets of cardinality one and cardinality two, an upper bound on the number of sets of cardinality three is established using shifting arguments. Then techniques from combinatorial design theory are used to establish a lower bound. Then it is shown that every number of sets of cardinality three between the lower and the upper bound can be realized. This characterization is restated to determine the precise spectrum of possible numbers of sets of cardinality two for specified numbers of sets of cardinality one and three. For simplicial complexes, these spectra are not always intervals, and the gaps are determined precisely. For simplicial multicomplexes, an alternative proof is given that these spectra are always intervals.