Erdős–Szekeres conjecture

Prove that ES(n)=2^{n-2}+1 for every integer n≥2.

Background

The Erdős–Szekeres conjecture concerns the minimum number of points in general position that guarantees n points in convex position. The survey states that the conjecture remains unresolved, although it is known for n≤6 and a near-sharp upper bound is available.

References

Based on their results for $n=2,3,4$, they posed the famous and still open Erd\H{os--Szekeres conjecture, for whose proof Erd\H{o}s offered \$500 reward.

— A Survey on Ordered Ramsey Numbers  (2502.02155 - Balko, 4 Feb 2025) in Conjecture~\ref{conj-intro-ES}, Subsection “Monotone Paths and the Erdős--Szekeres Theorem”

Based on their results for $n=2,3,4$, they posed the famous and still open Erd\H{os--Szekeres conjecture, for whose proof Erd\H{o}s offered \$500 reward.

— A Survey on Ordered Ramsey Numbers  (2502.02155 - Balko, 4 Feb 2025) in Conjecture conj-intro-ES, Section 3.1

Based on their results for $n=2,3,4$, they posed the famous and still open Erd\H{os--Szekeres conjecture, for whose proof Erd\H{o}s offered \$500 reward.

For every integer $n \geq 2$, we have ES(n) = 2{n-2}+1.

— A Survey on Ordered Ramsey Numbers  (2502.02155 - Balko, 4 Feb 2025) in Section 3, subsection “Monotone Paths and the Erdős–Szekeres Theorem,” Conjecture cited as [erdosSzekeres35]

Based on their results for $n=2,3,4$, they posed the famous and still open Erd\H{os--Szekeres conjecture, for whose proof Erd\H{o}s offered \$500 reward.

— A Survey on Ordered Ramsey Numbers  (2502.02155 - Balko, 4 Feb 2025) in Section 3, subsection “Monotone Paths and the Erdős–Szekeres Theorem”

Based on their results for $n=2,3,4$, they posed the famous and still open Erd\H{os--Szekeres conjecture, for whose proof Erd\H{o}s offered \$500 reward.

— A Survey on Ordered Ramsey Numbers  (2502.02155 - Balko, 4 Feb 2025) in Conjecture conj-intro-ES, subsection “Monotone Paths and the Erdős–Szekeres Theorem”