Speaker: Marta Pavelka

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 \emph{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

$$\text{ex}(n, S^d) \geq \Omega(n^{d + 1 – (d + 1)/(2^{d + 1} – 2)}).$$

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 $ex(n, S^d)$ 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. This is joint work with Andrew Newman.