Abstract
Let g
zs
(m, 2k) (g
zs
(m, 2k+1)) be the minimal integer such that for any coloring Δ of the integers from 1, . . . , g
zs
(m, 2k) by 
(the integers from 1 to g
zs
(m, 2k+1) by 
) there exist integers
such that
1. there exists j
x
such that Δ(x
i
) ∈ 
for each i and ∑
i
=1m Δ(x
i
) = 0 mod m (or Δ(x
i
)=∞ for each i);
2. there exists j
y
such that Δ(y
i
) ∈ 
for each i and ∑
i
=1m Δ(y
i
) = 0 mod m (or Δ(y
i
)=∞ for each i); and
1. 2(x m −x1)≤y m −x1.
In this note we show g
zs
(m, 2)=5m−4 for m≥2, g
zs
(m, 3)=7m+
−6 for m≥4, g
zs
(m, 4)=10m−9 for m≥3, and g
zs
(m, 5)=13m−2 for m≥2.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Baginsky, P.: A generalization of a Ramsey-type theorem on hypermatchings. Manuscript, 2002
Bialostocki, A.: Private Communication, June 2002
Bialostocki, A., Bialostocki, G., Schaal, D.: A zero-sum theorem. J. Combin. Theory, Ser A (to appear)
Bialostocki, A., Dierker, P.: On zero sum Ramsey numbers: multiple copies of a graph. J. Graph Theory 18, 143–151 (1994)
Bialostocki, A., Erdős P., Lefmann, H.: Monochromatic and zero-sum sets of nondecreasing diameter. Discrete Math. 137, 19–34 (1995)
Bialostocki, A., Sabar, R.: On constrained 2-Partitions of monochromatic sets and generalizations in the sense of Erdős-Ginzburg-Ziv. Manuscript, 2002
Bialostocki, A., Sabar, R., Schaal, D.: On a zero-sum generalization of a variation of Schur's equation. Preprint (2002)
Caro, Y.: Zero-Sum problems–a survey. Discrete Math. 152, 93–113 (1996)
Erdős, P., Ginzburg A., Ziv, A.: Theorem in additive number theory. Bull. Research Council Israel 10 F, 41–43 (1961)
Füredi, Z., Kleitman, D.: On zero-trees. J. Graph Theory 16, 107–120 (1992)
Grynkiewicz, D.: On a partition analog of the Cauchy-Davenport Theorem. Acta Math. Hungar. 107(1–2), 161–174 (2005)
Grynkiewicz, D.: On four colored sets with noncecreasing diameter and the Erdős-Ginzburg-Ziv Theorem. J. Combin. Theory, Ser. A 100, 44–60 (2002)
Grynkiewicz, D.: On four color monochromatic sets with nondecreasing diameter. Discrete Math. (to appear)
Grynkiewicz, D.: Sumsets, Zero-sums and Extremal Combinatorics. Ph.D. Dissertation, Caltech (2005)
Olson, J.E.: An addition theorem for finite abelian groups. J. Number Theorey 9, 63–70 (1977)
Sabar, R.: A note on a generalization in the sense of Erdős-Ginzburg-Ziv. Manuscript, 2002
Sabar, R.: Monochromatic and zero-sum solutions of a variation of Schur's equation. Manuscript, 2002
Sabar, R.: Partial-Ascending waves and the Erdő-s-Ginzburg-Ziv Theorem. Manuscript, 2002
Sabar, R.: On a family of inequalities and the Erdős-Ginzburg-Ziv Theorem. Manuscript, 2002
Schrijver, A., Seymour, P.: A simpler proof and a generalization of the zero-trees theorem. J. Combin. Theory Ser. A 58, 301–305 (1991)
Schultz, A.: On a modification of a problem of Bialostocki-Erdős-Lefmann. Discrete Math. 306, no. 2, 244–253 (2006)
Wong, B.: A generalization of the Bialostocki-Erdős-Lefmann theorem. Manuscript, 2002
Author information
Authors and Affiliations
Corresponding author
Additional information
Supported by NSF grant DMS 0097317
Rights and permissions
About this article
Cite this article
Grynkiewicz, D., Schultz, A. A Five Color Zero-Sum Generalization. Graphs and Combinatorics 22, 351–360 (2006). https://doi.org/10.1007/s00373-005-0636-x
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/s00373-005-0636-x