The $g$-conjecture (or $g$-theorem in the polytopal case) aims the characterize what possible $f$-vectors can arise from simplicial spheres. [Here][1] is some nice history and introduction to the $g$-conjecture by Gil Kalai. In brief the $g$-conjecture says the $g$-vector of a simplicial sphere should be an $M$-vector where $g_i = h_i - h_{i-1}$.


  [1]: https://gilkalai.wordpress.com/2009/04/04/how-the-g-conjecture-came-about/