# Barycentric-sum problem

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[Combinatorial number theory](/source/Combinatorial_number_theory) deals with [number theoretic](/source/number_theory) problems which involve [combinatorial](/source/combinatorics) ideas in their formulations or solutions. [Paul Erdős](/source/Paul_Erd%C5%91s) is the main founder of this branch of number theory. Typical topics include [covering system](/source/covering_system), [zero-sum problem](/source/zero-sum_problem)s, various [restricted sumset](/source/restricted_sumset)s, and [arithmetic progression](/source/arithmetic_progression)s in a set of integers. Algebraic or analytic methods are powerful in this field.

In combinatorial number theory, the '''barycentric-sum problems''' are questions that can be answered using combinatorial techniques. The context of barycentric-sum problems are the barycentric sequences.

==Example==

Let <math>Z_n</math> be the [cyclic group](/source/cyclic_group) of integers modulo ''n''. Let ''S'' be a sequence of elements of <math>Z_n</math>, where the repetition of elements is allowed. Let <math>|S|</math> be the length of ''S''. A sequence <math>S \subseteq Z_n</math> with <math>|S| \geq 2</math> is barycentric or has a
barycentric-sum if it contains one element <math>a_j</math> such that <math>\sum\limits_ {a_i \in S} a_i=|S|a_j</math>.

Informally, if <math>S</math> contains one element <math>a_j</math>, which is the ”average” of its terms. A barycentric sequence of length <math>t</math> is called a t-barycentric sequence. Moreover, when ''S'' is a set, the term barycentric set is used instead of barycentric sequence. For example, the set {0,1,2,3,4}  <math>\subseteq Z_8</math> is 5-barycentric with barycenter 2, however the set  {0,2,3,4,5} <math>\subseteq Z_8</math> is not 5-barycentric. The barycentric-sum problem consist in finding the smallest integer ''t'' such that any sequence of length ''t'' contains   a ''k''-barycentric sequence for some given ''k''. The study of the existence of such t related with k and the study of barycentric constants are part of the barycentric-sum problems. It has been introduced by Ordaz,<ref>C. Delorme, S. González, O. Ordaz and M.T. Varela. Barycentric sequences and barycentric Ramsey numbers stars, Discrete Math. 277(2004)45–56.</ref><ref>C. Delorme, I. Márquez, O. Ordaz and A. Ortuño. Existence condition
for barycentric sequences, Discrete Math. 281(2004)163–172.</ref> inspired in a theorem of Hamidoune:<ref>Y. O. Hamidoune. On weighted sequences sums, [Combinatorics, Probability and Computing](/source/Combinatorics%2C_Probability_and_Computing) 4(1995) 363–367.</ref> every sequence of length <math>n + k - 1 </math> in <math>Z_n</math> contains a k-barycentric sequence. Notice that a ''k''-barycentric sequence in <math>Z_n</math>, with k a multiple of n, is a sequence with zero-sum. The [zero-sum problem](/source/zero-sum_problem) on sequences started in 1961 with the Erdős, Ginzburg and Ziv theorem: every sequence of length <math>2n-1</math> in an [abelian group](/source/abelian_group) of order ''n'', contains an ''n''-subsequence with zero-sum.<ref>Y. Caro. Zero-sum problems: a survey. Discrete Math. 152 (1996) 93–113.</ref><ref>P. Erdős, A. Ginzburg and A. Ziv. Theorem in the additive number theory, Bull. Res. Council Israel 10F (1961) 41–43.</ref><ref>C. Flores and O. Ordaz. On sequences with zero sum in abelian group. Volume in homage to Dr. Rodolfo A. Ricabarra (Spanish), 99-106, Vol. Homenaje, 1, Univ. Nac. del Sur, Bahía Blanca, 1995.</ref><ref>W. Gao and A. Geroldinger, Zero-sum problems in finite abelian groups: A survey. Expositiones Mathematicae 24 (2006), n. 4, 337–369.</ref><ref>D. J. Grynkiewicz, O. Ordaz, M.T. Varela and F. Villarroel, On the Erd˝os-Ginzburg-Ziv Inverse Theorems. Acta Arithmetica. 129 (2007)307–318. 2</ref><ref>Y. O. Hamidoune, O. Ordaz and A. Ortuño. On a combinatorial theorem of Erdós-Ginzburg-Ziv. [Combinatorics, Probability and Computing](/source/Combinatorics%2C_Probability_and_Computing) 7 (1998)403–412.</ref><ref>O. Ordaz and D. Quiroz, Representation of group elements as subsequences
sums, To appear in Discrete Math.</ref>

Barycentric-sum problems have been defined in general for finite abelian groups. However, most of the main results obtained up to now are in <math>Z_n</math>.

The barycentric constants introduced by Ordaz are:<ref>S. González, L. González and O. Ordaz. Barycentric Ramsey numbers for small graphs, To appear in the Bulletin of the Malaysian Mathematical
Sciences Society.</ref><ref>L. González, I. Márquez, O. Ordaz and D. Quiroz, Constrained and generalized barycentric Davenport constants, Divulgaciones Matemáticas 15 No. 1 (2007)11–21.</ref><ref name="Barycentric Davenport constants">C. Guia, F. Losavio, O. Ordaz M.T. Varela and F. Villarroel, Barycentric Davenport constants. To appear in Divulgaciones Matemáticas.</ref><ref>O. Ordaz, M.T. Varela and F. Villarroel. k-barycentric Olson constant.
To appear in Mathematical Reports.</ref><ref>O. Ordaz and D. Quiroz, Barycentric-sum problem: a survey. Divulgaciones Matemáticas 15 No. 2 (2007)193–206.</ref> ''k''-barycentric Olson constant, ''k''-barycentric [Davenport constant](/source/Davenport_constant), barycentric Davenport constant, generalized barycentric Davenport constant, constrained barycentric Davenport constant. This constants are related to the Davenport constant<ref>C. Delorme, O. Ordaz and D. Quiroz. Some remarks on Davenport constant, Discrete Math. 237(2001)119–128.</ref> i.e. the smallest integer ''t'' such that any ''t''-sequence contains a [subsequence](/source/subsequence) with zero-sum. Moreover, related to the classical [Ramsey numbers](/source/Ramsey_numbers), the barycentric Ramsey numbers are introduced. An overview of the results computed manually or automatically are presented.<ref name="Barycentric Integers sequences">L. González, F. Losavio, O. Ordaz, M.T. Varela and F. Villarroel. Barycentric Integers sequences. Sumited to Expositiones Mathematicae.</ref> The implemented algorithms are written in C.<ref name="Barycentric Davenport constants"/><ref name="Barycentric Integers sequences"/><ref>F. Villarroel, Tesis Doctoral en Matemática. La constante de Olson k baricéntrica y un teorema inverso de Erdős-Ginzburg-Ziv. Facultad de Ciencias. Universidad Central de Venezuela, (2008).</ref>

==References==
<references/>

== External links ==
*[http://www.emis.de/journals/DM/ Divulgacions Matemáticas (Spanish)]

Category:Combinatorics

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