Optimality of the planar Erdős–Szekeres lower bound

Determine whether the lower bound ES_2(n) \geq 2^{n-2}+1 is asymptotically optimal for the planar Erdős–Szekeres convex polygon problem.

Background

The paper recalls that Erdős and Szekeres constructed 2{n-2} points in the plane containing no n points in convex position, yielding the lower bound ES_2(n) \geq 2{n-2}+1. The authors note that the bound is widely believed to be optimal, while the best known upper bound remains of the form 2{n+O(\sqrt{n\log n})}.

References

In 1960, they showed $ES_2 (n) \geq 2{n-2} +1$ by constructing $2{n-2}$ points in the plane containing no $n$ points in convex position and this bound is believed to be optimal.

— Big convex polytopes or rich hyperplanes  (2501.03645 - Furukawa, 7 Jan 2025) in Section 1.1, “Erdös-Szekeres theorem”