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Barycentric-sum problem

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Combinatorial number theory deals with number theoretic problems which involve combinatorial ideas in their formulations or solutions. Paul Erdős is the main founder of this branch of number theory. Typical topics include covering system, zero-sum problems, various restricted sumsets, and arithmetic progressions 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 Z n {\displaystyle Z_{n}} be the cyclic group of integers modulo n. Let S be a sequence of elements of Z n {\displaystyle Z_{n}} , where the repetition of elements is allowed. Let | S | {\displaystyle |S|} be the length of S. A sequence S ⊆ Z n {\displaystyle S\subseteq Z_{n}} with | S | ≥ 2 {\displaystyle |S|\geq 2} is barycentric or has a barycentric-sum if it contains one element a j {\displaystyle a_{j}} such that ∑ a i ∈ S a i = | S | a j {\displaystyle \sum \limits _{a_{i}\in S}a_{i}=|S|a_{j}} .

Informally, if S {\displaystyle S} contains one element a j {\displaystyle a_{j}} , which is the ”average” of its terms. A barycentric sequence of length t {\displaystyle t} 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} ⊆ Z 8 {\displaystyle \subseteq Z_{8}} is 5-barycentric with barycenter 2, however the set {0,2,3,4,5} ⊆ Z 8 {\displaystyle \subseteq Z_{8}} 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, inspired in a theorem of Hamidoune: every sequence of length n + k − 1 {\displaystyle n+k-1} in Z n {\displaystyle Z_{n}} contains a k-barycentric sequence. Notice that a k-barycentric sequence in Z n {\displaystyle Z_{n}} , with k a multiple of n, is a sequence with zero-sum. The zero-sum problem on sequences started in 1961 with the Erdős, Ginzburg and Ziv theorem: every sequence of length 2 n − 1 {\displaystyle 2n-1} in an abelian group of order n, contains an n-subsequence with zero-sum.

Barycentric-sum problems have been defined in general for finite abelian groups. However, most of the main results obtained up to now are in Z n {\displaystyle Z_{n}} .

The barycentric constants introduced by Ordaz are: k-barycentric Olson constant, k-barycentric Davenport constant, barycentric Davenport constant, generalized barycentric Davenport constant, constrained barycentric Davenport constant. This constants are related to the Davenport constant i.e. the smallest integer t such that any t-sequence contains a subsequence with zero-sum. Moreover, related to the classical Ramsey numbers, the barycentric Ramsey numbers are introduced. An overview of the results computed manually or automatically are presented. The implemented algorithms are written in C.

References

  1. C. Delorme, S. González, O. Ordaz and M.T. Varela. Barycentric sequences and barycentric Ramsey numbers stars, Discrete Math. 277(2004)45–56.
  2. C. Delorme, I. Márquez, O. Ordaz and A. Ortuño. Existence condition for barycentric sequences, Discrete Math. 281(2004)163–172.
  3. Y. O. Hamidoune. On weighted sequences sums, Combinatorics, Probability and Computing 4(1995) 363–367.
  4. Y. Caro. Zero-sum problems: a survey. Discrete Math. 152 (1996) 93–113.
  5. P. Erdős, A. Ginzburg and A. Ziv. Theorem in the additive number theory, Bull. Res. Council Israel 10F (1961) 41–43.
  6. 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.
  7. W. Gao and A. Geroldinger, Zero-sum problems in finite abelian groups: A survey. Expositiones Mathematicae 24 (2006), n. 4, 337–369.
  8. 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
  9. Y. O. Hamidoune, O. Ordaz and A. Ortuño. On a combinatorial theorem of Erdós-Ginzburg-Ziv. Combinatorics, Probability and Computing 7 (1998)403–412.
  10. O. Ordaz and D. Quiroz, Representation of group elements as subsequences sums, To appear in Discrete Math.
  11. S. González, L. González and O. Ordaz. Barycentric Ramsey numbers for small graphs, To appear in the Bulletin of the Malaysian Mathematical Sciences Society.
  12. L. González, I. Márquez, O. Ordaz and D. Quiroz, Constrained and generalized barycentric Davenport constants, Divulgaciones Matemáticas 15 No. 1 (2007)11–21.
  13. ^ C. Guia, F. Losavio, O. Ordaz M.T. Varela and F. Villarroel, Barycentric Davenport constants. To appear in Divulgaciones Matemáticas.
  14. O. Ordaz, M.T. Varela and F. Villarroel. k-barycentric Olson constant. To appear in Mathematical Reports.
  15. O. Ordaz and D. Quiroz, Barycentric-sum problem: a survey. Divulgaciones Matemáticas 15 No. 2 (2007)193–206.
  16. C. Delorme, O. Ordaz and D. Quiroz. Some remarks on Davenport constant, Discrete Math. 237(2001)119–128.
  17. ^ L. González, F. Losavio, O. Ordaz, M.T. Varela and F. Villarroel. Barycentric Integers sequences. Sumited to Expositiones Mathematicae.
  18. 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).

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