Erdős–Szekeres

Erdős–Szekeres
Name	Paul Erdős and George Szekeres problem
Caption	Paul Erdős and George Szekeres collaborated on combinatorial geometry
Birth date	1935 (paper)
Nationality	Hungarian-British (Erdős), Australian-Hungarian (Szekeres)
Fields	Combinatorics, Discrete geometry, Ramsey theory
Known for	Erdős–Szekeres theorem

Erdős–Szekeres

The Erdős–Szekeres problem originated in a 1935 combinatorial geometry note that linked ideas from Paul Erdős, George Szekeres, Ramsey theory, Combinatorics, Discrete geometry, and antecedent work by Moser family and contemporaries in Australasia and Europe. It asserts a structural inevitability in finite point sets and sparked decades of research by figures such as Van der Waerden, Graham, Ronald L., Endre Szemerédi, Pál Erdős, Paul Turán, and later contributors including János Pach, Miklós Simonovits, Imre Z. Ruzsa, and Béla Bollobás.

Statement and History

The original note by Paul Erdős and George Szekeres gave an elementary observation connecting ordered sequences and planar point configurations, echoing antecedents in the work of Erdős–Szekeres (1935), Folkman, and problems popularized by Richard Rado and Issai Schur. Early commentary linked the problem to combinatorial principles studied by Frank Ramsey and Péter Turán, and it influenced research programs at institutions like University of Cambridge, Princeton University, Bell Labs, University of Chicago, and Australian National University. Subsequent developments involved researchers from Massachusetts Institute of Technology, Universität Bonn, University of Szeged, University of Warwick, and ETH Zurich, with methods borrowing from techniques used by Paul Erdos collaborators such as Alfréd Rényi, Lajos Pósa, and Paul Turán.

Erdős–Szekeres Theorem (Happy Ending Problem)

The Erdős–Szekeres theorem, often called the "Happy Ending Problem" because the collaboration led to marriage between two participants in the social circle of the authors, states that for every positive integer n there exists a least integer ES(n) such that any ES(n) points in general position in the plane contain n points forming a convex n-gon. This statement intersects work by David Hilbert era geometers, Paul Erdős, George Szekeres, and later expansions by Ronald Graham, Endre Szemerédi, and János Pach. The theorem can be viewed as a planar instantiation of structural inevitability akin to results attributed to Frank Ramsey and resonates with combinatorial extremal problems studied at Institute for Advanced Study and Humboldt University of Berlin.

Bounds and Proofs

Erdős and Szekeres gave an upper bound ES(n) ≤ binomial(2n−4, n−2)+1 via a constructive combinatorial argument that used ordered sequences and pigeonhole principles akin to techniques by Erdős, Alfred Rényi, and Pál Turán. A classical lower bound arises from convex position constructions related to examples considered by Paul Erdős, George Szekeres, and later optimized by researchers at University of Cambridge and Princeton University including Ronald Graham and Béla Bollobás. Improvements and asymptotic refinements involve contributors such as János Pach, Miklós Simonovits, Imre Z. Ruzsa, Terence Tao, Ben Green, and Jacob Fox. Recent work lowering the coefficient in the asymptotic exponent used combinatorial geometry tools from groups including ETH Zurich and Columbia University teams; notable advances cite methods developed by Noga Alon, Zoltán Füredi, Gábor Tardos, and Márton Naszódi.

Proof techniques range from elementary combinatorial arguments familiar to students of Paul Erdős and George Szekeres to sophisticated methods invoking extremal set theory introduced by Péter Frankl, probabilistic techniques popularized by Béla Bollobás and Noga Alon, and geometric embedding ideas seen in work by János Pach and Miklós Simonovits. Constructive lower bounds often use configurations inspired by combinatorial constructions used by Erdős collaborators such as Alfréd Rényi and Jenő Lehel.

Generalizations and Variants

Generalizations address higher dimensions, colored versions, and order-type constraints. Higher-dimensional analogues link to problems studied at University of California, Berkeley and Stanford University by researchers like Gil Kalai, Jeff Kahn, and Jeff Erickson, exploring convex polytopes and Helly-type theorems connected to Helly and Carathéodory. Colored variants relate to Ramsey-type results advanced by Ronald Graham and Paul Erdős, while sequence analogues trace back to monotone subsequence results by I. J. Good and classical permutation pattern research influenced by Donald Knuth and Richard Stanley. Order-type and topological generalizations have been pursued by teams at University of Illinois Urbana–Champaign, Tel Aviv University, and University of Warsaw, including researchers Rade T. Živaljević, Ruiwen Xue, and Branko Grünbaum.

Applications and Related Results

The theorem and its extensions interact with computational geometry problems studied at Massachusetts Institute of Technology, Carnegie Mellon University, and Stanford University, influencing algorithms for convex hull computations by researchers such as Kirkpatrick and Preparata. Connections to Ramsey theory, extremal graph theory, and sequence combinatorics link the topic to work by Frank Ramsey, Paul Erdős, Ronald Graham, Endre Szemerédi, Béla Bollobás, and Noga Alon. Practical applications arise in pattern recognition research at Bell Labs and AT&T, in discrete optimization investigations at IBM Research, and in combinatorial design problems addressed at Los Alamos National Laboratory and University of Cambridge. Related classical results include the Erdős–Szekeres monotone subsequence theorem, Helly's theorem by Eduard Helly, and Carathéodory's theorem by Constantin Carathéodory, all of which have been taught and extended across institutions such as Princeton University, Harvard University, and University of Oxford.

Category:Erdős–Szekeres problem