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A conditional lower bound for the Turán number of spheres
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- Journal:
- Combinatorics, Probability and Computing , First View
- Published online by Cambridge University Press:
- 11 August 2025, pp. 1-9
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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.
