Optimality of the higher-dimensional lower bound

Establish whether the lower bound ES_d(n) \geq 2^{c n^{1/(d-1)}} is optimal for every dimension d \geq 2.

Background

For the higher-dimensional Erdős–Szekeres number ES_d(n), the paper recalls the construction of Károlyi and Valtr giving ES_d(n) \geq 2{c n{1/(d-1)}} for a dimension-dependent constant c>1. The authors explicitly state that the order of this lower bound is believed to be optimal, leaving the matching asymptotic upper bound unresolved.

References

On the other hand, K\n{a}rolyi and Valtr showed that there is a constant $c = c(d) > 1$ such that $ES_d (n) \geq 2{c n{\frac{1}{d-1}$ for every $d \geq 2$ and this bound is believed to be optimal.

— Big convex polytopes or rich hyperplanes  (2501.03645 - Furukawa, 7 Jan 2025) in Section 1.2, “Erdös-Szekeres theorem for higher dimension”