A conditional lower bound for the Turán number of spheres

Abstract We consider the hypergraph Turán problem of determining $ex(n, S^d)$ , 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 $S^d$ ) as a subcomplex. We show that if there is an affirmative answer to a question of Gromov about sphere enumeration in high dimensions, then $ex(n, S^d) \geq \Omega (n^{d + 1 - (d + 1)/(2^{d + 1} - 2)})$ . Furthermore, this lower bound holds unconditionally for 2-LC (locally constructible) spheres, which includes all shellable spheres and therefore all polytopes. We also prove an upper bound on $ex(n, S^d)$ 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.