For d≥3, let ES_d(n) be the least integer such that every set of ES_d(n) points in general position in R^d contains n points in convex position. Is ES_d(n)≥(1+c_d)^n for some constant c_d>0?

Erdős Problem #651 — Subexponential higher-dimensional Erdős–Szekeres numbers

About 29 years old · traced to 1997

For $d\geq3$ , let $ES_d(n)$ be the least integer such that every set of $ES_d(n)$ points in general position in $\mathbb{R}^d$ contains $n$ points in convex position. Is $ES_d(n)\geq(1+c_d)^n$ for some constant $c_d>0$ ?

References

Primary source

C. Pohoata and D. Zakharov, Convex polytopes from fewer points, arXiv:2208.04878 (2022).

Additional references

C. Pohoata and D. Zakharov, Convex polytopes from fewer points, arXiv:2208.04878 (2022).

convex geometry

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