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Published November 2006 | public
Journal Article

A Five Color Zero-Sum Generalization

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 ∪+^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.

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.

Additional details

Created:
August 22, 2023
Modified:
October 18, 2023