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        "description": "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.",
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      "statement": "# Erdős and Szekeres: monotone subsequences and unavoidable convex polygons\n\nA 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.\n\n## Scientific scope\n\nThe 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.\n\n## Reading status\n\nThis is a source-linked WRS catalogue note. A full explanatory guide and independent scientific reconstruction have not been completed for this entry. Read the identified original work for its complete narrative and arguments.\n\n## People and systems\n\nP. Erdős; G. Szekeres\n\nSource-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.\n\n\n\n## Original sources\n\n- [A combinatorial problem in geometry](https://www.numdam.org/item/CM_1935__2__463_0.pdf)\n",
      "summary": "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.",
      "title": "Erdős and Szekeres: monotone subsequences and unavoidable convex polygons"
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    "title": "Erdős and Szekeres: monotone subsequences and unavoidable convex polygons",
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