A conditional lower bound for the Turán number of spheres

Abstract

We consider the hypergraph Turán problem of determining $\mathrm{ex}(n,S^d)$, 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 $\mathrm{ex}(n,S^d)\ge \Omega (n^{d+1-(d+1)/(2^{d+1}-2)})$. 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 $\mathrm{ex}(n,S^d)$ 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.

9 pages

References in corpus (1)

• 2-LC triangulated manifolds are exponentially many