SWhat changed
A long enough ordered list of distinct numbers must contain a long increasing or decreasing subsequence, even if that pattern is hidden among other entries. Erdős and Szekeres establish the sharp threshold (r−1)(s−1)+1 for an increasing subsequence of length r or a decreasing one of length s. Their related geometric argument shows that sufficiently many planar points in general position contain a convex n-gon, with the historical upper bound binomial(2n−4,n−2)+1.
Scientific scope
A long enough ordered list of distinct numbers must contain a long increasing or decreasing subsequence, even if that pattern is hidden among other entries. Erdős and Szekeres establish the sharp threshold (r−1)(s−1)+1 for an increasing subsequence of length r or a decreasing one of length s. Their related geometric argument shows that sufficiently many planar points in general position contain a convex n-gon, with the historical upper bound binomial(2n−4,n−2)+1. The strictly convex polygon statement is taken in general position, with no three points collinear; the paper separately discusses allowing collinear boundary vertices. The binomial polygon bound is an upper bound, not the exact minimum, and the proposed 2^(n−2)+1 value is a conjecture in this source. The monotone-subsequence threshold, unlike the polygon estimate, is sharp under the stated distinctness convention. Higher-dimensional extensions and every later quantitative improvement are not assessed.
Evidence
The linked original works support the scoped statement and contribution attribution. WRS has not independently replayed experiments or reconstructed a full mathematical proof. The strictly convex polygon statement is taken in general position, with no three points collinear; the paper separately discusses allowing collinear boundary vertices. The binomial polygon bound is an upper bound, not the exact minimum, and the proposed 2^(n−2)+1 value is a conjecture in this source. The monotone-subsequence threshold, unlike the polygon estimate, is sharp under the stated distinctness convention. Higher-dimensional extensions and every later quantitative improvement are not assessed.
Contributors
- The named authors in the research record · SCIENCE: Source-listed authors for the exact contribution; experimental subjects are not autonomous scientific authors. Source support and predecessor work retain separate roles. Individual effort shares and unresolved identities are not inferred.
- Esther Klein · SCIENCE: Posed the convex-polygon problem and supplied the five-point argument reproduced in the introduction.
- E. Makai · SCIENCE: The introduction credits the exact five-gon threshold of nine points.
- Richard Rado · SCIENCE: Independently considered the monotone-subsequence problem, as the original footnote records.
- D. König · SCIENCE: Earlier infinity lemma supplies an alternative existence argument cited by the authors.
Provisional WRS assessment, edition 2026-10-23.4. Independent expert calibration is pending. This grade does not certify correctness or assign ownership.
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