Erdős–Szekeres conjecture on convex-position numbers

Prove that for every integer k ≥ 3, g(k) = 2^{k−2} + 1, where g(k) denotes the smallest integer such that any set of g(k) points in the plane in general position (with no three collinear) contains k points in convex position.

Background

The Erdős–Szekeres problem asks for the minimum number g(k) such that any planar point set in general position of size g(k) contains k points in convex position. The exact values are known only for small k: g(4)=5, g(5)=9, and g(6)=17. Erdős and Szekeres proposed the formula g(k)=2^{k−2}+1 and proved it as a lower bound, but the general case remains unproven. In this work, the authors use SAT-based encodings with rotational symmetry to find and classify symmetric extremal configurations for related instances, providing computational evidence and tools but not resolving the conjecture itself.