In mathematics, zero-sum Ramsey theory or zero-sum theory is a branch of combinatorics.
It deals with problems of the following kind: given a combinatorial structure whose elements are assigned different weights (usually elements from an Abelian group
), one seeks for conditions that guarantee the existence of certain substructure whose weights of its elements sum up to zero (in
It combines tools from number theory, algebra, linear algebra, graph theory, discrete analysis, and other branches of mathematics.
The classic result in this area is the 1961 theorem of Paul Erdős, Abraham Ginzburg, and Abraham Ziv:[1] for any
[2] (This bound is tight, as a sequence of
[2] Generalizing this result, one can define for any abelian group G the minimum quantity
is the order of the group) which adds to zero.
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