Title

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 ∪+ i=1κ ℤ mi (the integers from 1 to g zs (m, 2k+1) by ∪+ i=1κ ℤ mi ∪ {∞}) there exist integers x 1 <...< x m < y 1 <...< y m such that 1. there exists j x such that Δ(x i ) ∈ ℤ mjx 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 ) ∈ ℤ mjy for each i and ∑ i=1m Δ(y i ) = 0 mod m (or Δ(y i )=∞ for each i); and 1. 2(x m -x 1) ≤ y m -x 1. 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. © Springer-Verlag Berlin Heidelberg 2006.

Publication Title

Graphs and Combinatorics

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