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A Five Color Zero-Sum Generalization

Grynkiewicz, David and Schultz, Andrew (2006) A Five Color Zero-Sum Generalization. Graphs and Combinatorics, 22 (3). pp. 351-360. ISSN 0911-0119. doi:10.1007/s00373-005-0636-x.

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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 ∪+^k_(i=1)Z^i_m (the integers from 1 to g_(zs) (m, 2k+1) by ∪+^k_(i=1) Z^i_m ∪ {∞ }) there exist integers x₁ < … < x_m < y₁ < … y_m such that 1. there exists j_x such that Δ(x_i) ∈ Z^(jx)_m for each i and ∑_(i =1)^m Δ(x_i) = 0 mod m (or Δ(x_i) = ∞ for each i); 2. there exists j_y such that Δ(y i) ∈ Z^(jy)_m for each i and ∑_(i =1)^m 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+[m/2]−6 for m ≥ 4, g_(zs) (m, 4) = 10m−9 for m ≥ 3, and g_(zs) (m, 5) = 13m−2 for m ≥ 2.

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Additional Information:© 2006 Springer-Verlag. Received: October 10, 2002; Final version received: September 21, 2005. Supported by NSF grant DMS 0097317. The authors would like to thank Professor A. Bialostocki for suggesting that we investigate Conjectures 1.3 and for many fruitful discussions, and the referees for their useful suggestions.
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Subject Keywords:Discrete Math; Trivial Fact; Ramsey Number; Positive Integer Solution; Additive Number Theory
Issue or Number:3
Record Number:CaltechAUTHORS:20200103-150222668
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Official Citation:Grynkiewicz, D., Schultz, A. A Five Color Zero-Sum Generalization. Graphs and Combinatorics 22, 351–360 (2006) doi:10.1007/s00373-005-0636-x
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:100513
Deposited By: Tony Diaz
Deposited On:05 Jan 2020 03:45
Last Modified:16 Nov 2021 17:54

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