The Erdős–Szekeres Convex Polygon Problem (Exact Values)

Define ES(n) = minimum N such that any set of N points in general position contains n in convex position. Known: ES(3)=3, ES(4)=5, ES(5)=9, ES(6)=17. Open: determine ES(n) for n ≥ 7. Erdős–Szekeres conjectured ES(n) = \(2^{n−2}\) + 1.

Erdős and Szekeres founded the field in their 1935 paper proving finiteness. The upper bound stood at ~\(4^{n}\) for sixty years until Suk's polynomial-method breakthrough achieved \(2^{n+o(n)}\). Holmsen, Mojarrad and others improved lower-bound constructions. Computer verification confirms ES(7) = ? between 33 and 127 — even this next step is computationally infeasible by brute force. The problem birthed modern Ramsey-type combinatorial geometry.