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September 10, 2025Combinatorics Probability Computing

A conditional lower bound for the Turán number of spheres

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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.

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Implication

The study shows a lower bound on Turán numbers in simplicial complexes, revealing implications for locally constructible spheres and conjecturing improvements on upper bounds.

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Newman et al. (2025) studied this question.

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

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Implication

The study shows a lower bound on Turán numbers in simplicial complexes, revealing implications for locally constructible spheres and conjecturing improvements on upper bounds.

A conditional lower bound for the Turán number of spheres

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Implication

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A conditional lower bound for the Turán number of spheres

Key Points

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Implication

Cite This Study

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