Erdos–Szekeres conjecture on points in convex position

Determine whether for every integer n ≥ 3, any set of at least 2^(n−2)+1 points in the Euclidean plane with no three collinear necessarily contains n points in convex position; equivalently, determine the exact value of N(n), the minimal number such that any set of N(n) planar points in general position contains n points in convex position (the Erdos–Szekeres conjecture).

Background

The paper studies necessary and sufficient conditions under which a set of n distinct planar points P1(x1,y1), ..., Pn(xn,yn) with strictly increasing x-coordinates forms a convex n-gon, connecting properties of sequential convexity with convex polygon structure. In motivating this line of inquiry within discrete geometry, the authors cite classical themes such as Radon’s, Helly’s, and Carathéodory’s results, and they explicitly reference a prominent unresolved problem.

The Erdos–Szekeres conjecture concerns guaranteeing the existence of n points in convex position within sufficiently large point sets in general position. While the present work provides deterministic coordinate-wise conditions to ensure convexity of polygons formed by ordered points, the conjecture addresses existence without such structural assumptions, underscoring a broader open direction in the area.