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September 10, 2025

A conditional lower bound for the Turán number of spheres

Key Points

A new lower bound for the Turán number of spheres has been established, enhancing existing understanding of simplicial complexes.
This lower bound is at least $A9(n^{d + 1 - (d + 1)/(2^{d + 1} - 2)})$, depending on a high-dimensional question posed by Gromov.
An upper bound on the Turán number has also been derived, being $O(n^{d + 1 - 1/2^{d - 1}})$, indicating possible future improvements.
The findings help clarify relationships between shellable spheres, polytopes, and Turán numbers in the structure of high-dimensional spaces.

Abstract

Abstract We consider the hypergraph Turán 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 (locally constructible) 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.

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Cite This Study

Newman et al. (Mon,) studied this question.

synapsesocial.com/papers/68c1c31254b1d3bfb60f052c https://doi.org/https://doi.org/10.1017/s0963548325100096

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