A conditional lower bound for the Tur\'an number of spheres

We consider the hypergraph Tur\'an problem of determining ex (n, Sᵈ), the maximum number of facets in a d-dimensional simplicial complex on n vertices that does not contain a simplicial d-sphere (a homeomorph of Sᵈ) as a subcomplex. We show that if there is an affirmative answer to a question of Gromov about sphere enumeration in high dimensions, then ex (n, Sᵈ) (n^d + 1 - (d + 1) / (2^{d + 1 - 2) }). Furthermore, this lower bound holds unconditionally for 2-LC spheres, which includes all shellable spheres and therefore all polytopes. We also prove an upper bound on ex (n, Sᵈ) of O (n^d + 1 - 1/2^{d - 1}) using a simple induction argument. We conjecture that the upper bound can be improved to match the conditional lower bound.