Abstract
Let \(m\geqslant 3\) be a positive integer, and let \(\mathbb{Z}_m\) denote the cyclic group of residues modulo m. Furthermore, let \(R(L_m; 2)(R(L_m;\mathbb{Z}_m))\) denote the minimum integer N such that for every function \(\Delta: \{1,2,\ldots,N\}\rightarrow \{0,1\}\,(\Delta: \{1,2,\ldots,N\}\rightarrow \mathbb{Z}_m)\) there exist m integers \(x_1< x_2<\cdots< x_m\) satisfying \(\sum_{i=1}^{m-1}x_i< x_m\) and \(\Delta(x_1)=\Delta(x_2)=\cdots=\Delta(x_m)\) (and \(\sum_{i=1}^{m}\Delta(x_i)=0\)). It is shown that \(R(L_{m};2)=R(L_m;\mathbb{Z}_m)\) for every odd prime m.
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References
Baginski, P.: A generalization of a Ramsey-type theorem on hypermatchings. J. Graph Theory 50(2), 142–149 (2005)
Beutelspacher, A., Brestovansky, W: Generalized Schur numbers. Lect. Notes Math., vol. 969, 30–38 (1982)
Bialostocki, A.: Zero sum trees: a survey of results and open problems. Finite and infinite combinatorics in sets and logic (Banff, AB, 1991), 19-29, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 411, Kluwer Acad. Publ., Dordrecht, 1993
Bialostocki, A., Bialostocki, G., Schaal, D.: A zero-sum theorem. J. Combin. Theory Ser. A 101(1), 147–152 (2003)
Bialostocki, A., Dierker, P.: Zero sum Ramsey theorems. Proceedings of the Twentieth Southeastern Conference on Combinatorics, Graph Theory, and Computing (Boca Raton, FL, 1989). Congr. Numer. 70, 119–130 (1990)
Bialostocki, A., Dierker, P.: On zero sum Ramsey numbers: small graphs. Twelfth British Combinatorial Conference (Norwich, 1989). Ars Combin. 29 A, 193–198 (1990)
Bialostocki, A., Dierker, P.: On the Erdős-Ginzburg-Ziv theorem and the Ramsey numbers for stars and matchings. Discrete Math. 110(1–3), 1–8 (1992)
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. Ars Combin. 76, 277–286 (2005)
Bialostocki, A., Schaal, D.: On a variation of Schur numbers. Graphs Combin. 16(2), 139–147 (2000)
Caro, Y.: Zero-sum problems-a survey. Discrete Math. 152(1–3), 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.J.: On four colored sets with nondecreasing diameter and the Erdős-Ginzburg-Ziv Theorem. J. Combin. Theory Ser. A 100(1), 44–60 (2002)
Grynkiewicz, D.J., Sabar, R.: Monochromatic and zero-sum sets of nondecreasing modified diameter. Electron. J. Combin. 13 (2006), no. 1, Research Paper 28, 19 pp. (electronic)
Grynkiewicz, D.J., Schultz, A.: A five color zero-sum generalization. Graphs Combin.22(3), 351–360 (2006)
Schaal, D.: On some zero-sum Rado type problems. (Ph.D. dissertation, 1994), University of Idaho, Moscow Idaho, USA
Schrijver, A., Seymour, P.: A simpler proof and a generalization of the zero-trees theorem. J. Combin. Theory Ser. A 58, 301–305 (1991)
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Rasheed Sabar: Conducted the research at an REU program at the University of Idaho with the support of NSF grant DMS0097317.
Daniel Schaal: Partially supported by a South Dakota Governor’s 2010 Individual Research Seed Grant.
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Bialostocki, A., Sabar, R. & Schaal, D. On a Zero-Sum Generalization of a Variation of Schur’s Equation. Graphs and Combinatorics 24, 511–518 (2008). https://doi.org/10.1007/s00373-008-0815-7
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DOI: https://doi.org/10.1007/s00373-008-0815-7