What does this research mean for the field?

The foundational results of the Erdős-Szekeres Convex Polygon Theorem, including Esther Klein's theorem and the Cup-Cap duality, can be rigorously exposited and formally verified using the Lean 4 interactive theorem prover. Novelty: ClaimNovelty.METHODOLOGICAL. Consensus alignment: ConsensusAlignment.NEUTRAL.

What question did this study set out to answer?

To provide a comprehensive pedagogical treatise on the Erdős-Szekeres Convex Polygon Theorem alongside a 100% machine-checked formal verification in Lean 4.

August 29, 2026Open Access

On the Erdős-Szekeres Convex Polygon Theorem

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CNCharles EDOU NZE

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Implication

Theoretical analysis demonstrates complete formal proofs of the Erdős-Szekeres convex polygon theorem in Lean 4, highlighting machine-checked bounds in geometric Ramsey theory.

Key Points

To provide a comprehensive pedagogical treatise on the Erdős-Szekeres Convex Polygon Theorem alongside a 100% machine-checked formal verification in Lean 4.
Synthesized analytic proofs for Esther Klein's theorem, the classical Cup-Cap Ramsey recurrence, and Andrew Suk's 2017 asymptotic bound using hypergraph Ramsey regularity.
Constructed explicit algebraic counterexamples demonstrating the strict lower bound for four-point convex configurations.
Formally encoded general position predicates, orientation determinants, and combinatorial identities using the Lean 4 interactive theorem prover and Mathlib with zero axioms or sorry placeholders.
Derived the step-by-step partition proof establishing g(4) = 5 via convex hull cardinality analysis, confirming g(4) > 4 with an explicit triangle containing one interior point.
Rigorously proved 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 alongside Suk's asymptotic bound g(n) = 2^(n + O(n^(2/3) log n)).
Achieved complete, error-free machine verification in Lean 4 for the discrete orientation predicates, the Cup-Cap binomial identity, and the exact value g(4) = 5.

Cite This Study

Charles EDOU NZE (2026) studied this question.

synapsesocial.com/papers/6a92994d8e5d7d1fc0c112c8 https://doi.org/10.5281/zenodo.22132974

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Implication

Theoretical analysis demonstrates complete formal proofs of the Erdős-Szekeres convex polygon theorem in Lean 4, highlighting machine-checked bounds in geometric Ramsey theory.

On the Erdős-Szekeres Convex Polygon Theorem

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Implication

Key Points

Cite This Study

On the Erdős-Szekeres Convex Polygon Theorem

Discussion

Member takes

Implication

Key Points

Cite This Study

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