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| Erdős–Ginzburg–Ziv theorem | |
|---|---|
| Name | Erdős–Ginzburg–Ziv theorem |
| Field | Combinatorics |
| Authors | Paul Erdős; Abraham Ginzburg; Abraham Ziv |
| Year | 1961 |
| Statement | For any 2n−1 integers there exist n whose sum is divisible by n |
Erdős–Ginzburg–Ziv theorem is a result in Paul Erdős-era combinatorial number theory first proved by Paul Erdős, Abraham Ginzburg, and Abraham Ziv in 1961, asserting a zero-sum property for multisets of integers modulo n. The theorem sits within the tradition of additive combinatorics associated with figures such as Harald Cramér, Paul Turán, Erdős–Turán conjecture, Ramsey theory, and Olga Taussky-Todd, and has influenced subsequent work by Richard Rado, Kurt Mahler, Jean-Pierre Serre, and Endre Szemerédi.
For a positive integer n and any multiset of 2n−1 integers there exists a submultiset of size n whose sum is divisible by n; this formulation connects to results of James Joseph Sylvester, Isaac Newton-era congruence reasoning, Carl Friedrich Gauss's modular arithmetic, and later modular combinatorics studied by Andrey Kolmogorov, Kurt Gödel, and John von Neumann. The theorem is often phrased in the language of finite abelian groups following work by Emmy Noether, Niels Henrik Abel, Évariste Galois, David Hilbert, and Emil Artin that situates the result for cyclic groups of order n and links to the Davenport constant studied by Harald Davenport, Antonín Novák, Alfred Rényi, and George Pólya.
The original 1961 paper by Paul Erdős, Abraham Ginzburg, and Abraham Ziv built on combinatorial number theory traditions traced through Srinivasa Ramanujan, G. H. Hardy, John Littlewood, and Paul Erdős's extensive collaborations with Paul Turán and Béla Bollobás. Subsequent rediscoveries and expositions involved Pál Erdős-era seminars and were influenced by work of Richard Dedekind, Leopold Kronecker, L. Carlitz, Hans Rademacher, and I. M. Gelfand; notable early commentary appeared in surveys by Miklós Schweitzer competition-affiliated authors and in expositions connected to International Mathematical Olympiad training that referenced Andrews-type partition identities. The theorem entered textbooks alongside contributions by Israel Gelfand, Paul Halmos, Norbert Wiener, and E. T. Bell.
Classical combinatorial proofs employ the pigeonhole principle in the vein of Srinivasa Ramanujan-style partition arguments, polynomial method influences from Noga Alon and Paul Erdős, and group-theoretic reductions traceable to Évariste Galois and Emmy Noether. Alternative proofs use zero-sum theory developed by Harald Davenport and refined by Graham H. Hardy-school combinatorialists, while algebraic-combinatorial approaches exploit Chevalley–Warning-type techniques linked to Claude Chevalley and E. Warning; further analytic perspectives invoke Fourier-analytic tools associated with Jean Bourgain, Terence Tao, Ben Green, Timothy Gowers, and Endre Szemerédi. Short, elegant proofs have been given in expository works by Paul Erdős collaborators such as Endre Szemerédi, Imre Z. Ruzsa, Ronald Graham, and Melvin Olson.
Generalizations extend to arbitrary finite abelian groups via the Davenport constant and Olson's constant, with landmark contributions by Károly Bezdek, Gao Xianchang, Alfred Geroldinger, Wolfgang A. Schmid, and Peter van Emde Boas. Higher-multiplicity and weighted variants connect to results by Noga Alon, Miklós Bóna, János Pach, Rudolf Ahlswede, and Vojtěch Rödl; multidimensional extensions link to the Erdős–Ginzburg–Ziv-type constants studied by Richard Stanley, Kurt Mahler, Jean-Pierre Serre, and László Lovász. Related theorems include the Erdős–Heilbronn conjecture resolved by D. J. Newman methods, the Olson theorem, and zero-sum Ramsey-type statements associated with Frank Ramsey and Richard Rado.
Concrete applications appear in combinatorial designs studied by Raymond Paley, Kurt Gödel-adjacent combinatorics, and additive bases research by Paul Erdős and Sidney Newman; coding-theoretic connections invoke Claude Shannon, Richard Hamming, and Elwyn Berlekamp. Examples include explicit constructions for n=2,3,5 illustrated using small cases familiar from International Mathematical Olympiad problem sets and techniques attributed to G. H. Hardy and John Littlewood; algorithmic implications have been explored in complexity contexts linked to Stephen Cook, Richard Karp, and Leslie Valiant.
Ongoing research centers on exact values of generalized zero-sum constants for finite abelian groups with contributions from Alfred Geroldinger, Wolfgang A. Schmid, Gao Xianchang, and W. D. Gao, and on computational complexity questions influenced by Stephen Cook, László Babai, and Avi Wigderson. Other directions probe weighted versions and asymptotic bounds inspired by Ben Green, Terence Tao, Timothy Gowers, and József Beck, and seek structural characterizations reminiscent of inverse problems studied by Paul Erdős, Endre Szemerédi, Miklós Z. Ruzsa, and Imre Z. Ruzsa.