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On a zero-sum generalization of a variation of Schur’s equation. (English) Zbl 1163.05055

In this paper the authors study a variant of Schur’s problem: Let \(m\geq 3\) be a positive integer. Let \(R(L_m;2)\) (or \(R(L_m; \mathbb{Z}_m\)) respectively) denote the minimum integer \(N\) such that for every function \(\Delta:\{1,2, \ldots , N\} \rightarrow \{0,1\}\) (or \(\Delta:\{1,2, \ldots , N\} \rightarrow \mathbb{Z}_m\)) there exist \(m\) integers \(x_1<x_2 < \cdots < x_m\) with \(\sum_{i=1}^{m-1} x_i <x_m\) and \(\Delta(x_1)=\Delta(x_2)= \cdots =\Delta(x_m)\) (and \(\sum_{i=1}^m \Delta(x_i)=0\)). In this paper it is proved that \(R(L_m;2)=R(L_m;\mathbb{Z}_m)\), for every odd prime \(m\). An explicit value of \(R(L_m;2)\) had been worked out in [A. Bialostocki and D. Schaal, ”On a variation of Schur numbers,” Graphs Comb. 16, No.2, 139-147 (2000; Zbl 0973.05080)].

MSC:

05D10 Ramsey theory
11B50 Sequences (mod \(m\))

Citations:

Zbl 0973.05080
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References:

[1] Baginski, P.: A generalization of a Ramsey-type theorem on hypermatchings. J. Graph Theory 50(2), 142–149 (2005) · Zbl 1099.05059
[2] Beutelspacher, A., Brestovansky, W: Generalized Schur numbers. Lect. Notes Math., vol. 969, 30–38 (1982) · Zbl 0498.05002
[3] 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 · Zbl 0845.05069
[4] Bialostocki, A., Bialostocki, G., Schaal, D.: A zero-sum theorem. J. Combin. Theory Ser. A 101(1), 147–152 (2003) · Zbl 1017.05102
[5] 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) · Zbl 0695.05050
[6] Bialostocki, A., Dierker, P.: On zero sum Ramsey numbers: small graphs. Twelfth British Combinatorial Conference (Norwich, 1989). Ars Combin. 29 A, 193–198 (1990) · Zbl 0708.05039
[7] Bialostocki, A., Dierker, P.: On the Erdos-Ginzburg-Ziv theorem and the Ramsey numbers for stars and matchings. Discrete Math. 110(1–3), 1–8 (1992) · Zbl 0774.05065
[8] Bialostocki, A., Dierker, P.: On zero sum Ramsey numbers: multiple copies of a graph. J. Graph Theory 18, 143–151 (1994) · Zbl 0794.05081
[9] Bialostocki, A., Erdos, P., Lefmann, H.: Monochromatic and zero-sum sets of nondecreasing diameter. Discrete Math. 137, 19–34 (1995) · Zbl 0822.05046
[10] Bialostocki, A., Sabar, R.: On constrained 2-partitions of monochromatic sets and generalizations in the sense of Erdos-Ginzburg-Ziv. Ars Combin. 76, 277–286 (2005) · Zbl 1164.05462
[11] Bialostocki, A., Schaal, D.: On a variation of Schur numbers. Graphs Combin. 16(2), 139–147 (2000) · Zbl 0973.05080
[12] Caro, Y.: Zero-sum problems-a survey. Discrete Math. 152(1–3), 93–113 (1996) · Zbl 0856.05068
[13] Erdos, P., Ginzburg, A., Ziv, A.: Theorem in additive number theory. Bull. Research Council Israel 10 F, 41–43 (1961)
[14] Füredi, Z., Kleitman, D.: On zero-trees. J. Graph Theory 16, 107–120 (1992) · Zbl 0772.05033
[15] Grynkiewicz, D.J.: On four colored sets with nondecreasing diameter and the Erdos-Ginzburg-Ziv Theorem. J. Combin. Theory Ser. A 100(1), 44–60 (2002) · Zbl 1027.11016
[16] 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) · Zbl 1084.05073
[17] Grynkiewicz, D.J., Schultz, A.: A five color zero-sum generalization. Graphs Combin.22(3), 351–360 (2006) · Zbl 1111.05095
[18] Schaal, D.: On some zero-sum Rado type problems. (Ph.D. dissertation, 1994), University of Idaho, Moscow Idaho, USA
[19] Schrijver, A., Seymour, P.: A simpler proof and a generalization of the zero-trees theorem. J. Combin. Theory Ser. A 58, 301–305 (1991) · Zbl 0756.05085
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