On the Erdős-Szekeres Convex Polygon Theorem

This preprint provides an exhaustive mathematical treatise on the Erdős-Szekeres Convex Polygon Theorem (Problem #08 in Paul Erdős' collection, historically known as the "Happy Ending Problem"), a foundational cornerstone of discrete geometry and geometric Ramsey theory. Originating from a 1933 observation by Esther Klein and formalized in the seminal 1935 paper by Paul Erdős and George Szekeres, the theorem asserts that every sufficiently large set of points in the Euclidean plane in general position contains a subset of n points in convex position, with the exact threshold conjectured to be g(n) = 2n−2 + 1: g(n) = 2n−2 + 1 &implies; ∃ S ⊆ P, |S| = n in convex position This treatise provides a non-elliptical exposition of Esther Klein's theorem (g(4) = 5), the classical Cup-Cap Theorem N(m, ℓ) = C(m + ℓ − 4, m − 2) + 1, and Andrew Suk's landmark 2017 proof establishing the near-tight asymptotic upper bound g(n) = 2n + O(n2/3 log n) via hypergraph Ramsey regularity. Key Mathematical Results & Contributions Complete Proof of Esther Klein's Theorem: Step-by-step partition proof establishing g(4) = 5 via convex hull cardinality analysis (|conv(P)| ∈ {3, 4, 5}) and interior diagonal triangulation. Exact 4-Point Non-Convex Counterexample: Algebraic demonstration that g(4) > 4 via an explicit triangle with one strictly interior point. The Cup-Cap Ramsey Duality Theorem: Rigorous inductive derivation of the bivariate recurrence N(m, ℓ) = N(m − 1, ℓ) + N(m, ℓ − 1) − 1, establishing the classical upper bound g(n) ≤ C(2n − 4, n − 2) + 1. Andrew Suk's Breakthrough (2017): Pedagogical overview of positive-fraction hypergraph Ramsey theorems and the reduction of the error term in g(n). 100% Machine-Checked Verification in Lean 4: General position predicates, discrete orientation determinants, the exact value g(4) = 5, and the Cup-Cap binomial identity are formally verified with 0 axioms, 0 linter warnings, and 0 sorry placeholders via the Lean 4 interactive theorem prover and Mathlib. Repository and Verification Artifacts The companion machine-checked code and formal verification artifacts are publicly hosted on GitHub:https://github.com/flouzzy/erdos-problemsInteractive Portal: https://maths-proofs.edounze.com/#erdos-08-szekeres Primary MSC (2020): 52C10, 05D10, 68V20, 52A10, 05A17.Keywords: Erdős-Szekeres Theorem, Happy Ending Problem, Combinatorial Geometry, Ramsey Theory, Cup-Cap Duality, Convex Polygons, Formal Verification, Lean 4, Mathlib.