The number of k-faces of a simple d-polytope

Consider the question: Given integers 0 less than or equal to k < d < n, does there exist a simple d-polytope with n faces of dimension k? We show that there exist numbers G(d, k) and N(d, k) such that for n > N(d, k) the answer is yes if and only if G(d, k) divides n, Furthermore, a formula for G(d, k) is given, showing that, e.g., G(d, k) = 1 if k greater than or equal to [(d + 1)/2] or if both d and k are even, and also in some other cases (meaning that all numbers beyond N(d, k) occur as the number of k-faces of some simple d-polytope). This question has previously been studied only for the case of vertices (k = 0), where Lee [Le] proved the existence of N(d, 0) (with G(d, 0) = 1 or 2 depending on whether d is even or odd), and Prabhu [P1] showed that N(d, 0) less than or equal to cd root d. We show here that asymptotically the true value of Prabhu's constant is c = root 2 if d is even, and c = 1 if d is odd.