One important open problem in the field was the g-conjecture, formulated by Peter McMullen, which asks about possible numbers of faces of different dimensions of a simplicial sphere.
In December 2018, the g-conjecture was proven by Karim Adiprasito in the more general context of rational homology spheres.
By repeatedly performing the barycentric subdivision, it is easy to construct a simplicial sphere for any n ≥ 4.
The g-conjecture, formulated by McMullen in 1970, asks for a complete characterization of f-vectors of simplicial d-spheres.
In the case of polytopal spheres, the answer is given by the g-theorem, proved in 1979 by Billera and Lee (existence) and Stanley (necessity).