What is recorded here?
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.
Source-linked catalogue note. A detailed explanatory guide has not yet been prepared for this entry.
Scientific scope
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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People and systems
P. Erdős; G. Szekeres
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
Posed the convex-polygon problem and supplied the five-point argument reproduced in the introduction.
E. Makai
The introduction credits the exact five-gon threshold of nine points.
Richard Rado
Independently considered the monotone-subsequence problem, as the original footnote records.
D. König
Earlier infinity lemma supplies an alternative existence argument cited by the authors.
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English catalogue narrative
Erdős and Szekeres: monotone subsequences and unavoidable convex polygons
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
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.
Reading status
This 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.
People and systems
P. Erdős; G. Szekeres
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.