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A Five Color Zero-Sum Generalization
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A Five Color Zero-Sum Generalization
David Grynkiewicz1  and Andrew Schultz2
| (1) |
Department of Mathematics, Caltech, Pasadena, CA, 91125 |
| (2) |
Department of Mathematics, Stanford University, Stanford, CA, 94305 |
Received: 10 October 2002 Accepted: 21 September 2005
Abstract Let gzs( m, 2 k) ( gzs( m, 2 k+1)) be the minimal integer such that for any coloring Δ of the integers from 1, . . . , gzs( m, 2 k) by  (the integers from 1 to gzs( m, 2 k+1) by  ) there exist integers
1. there exists jx such that Δ( xi) ∈  for each i and ∑ i=1m Δ( xi) = 0 mod m (or Δ( xi)=∞ for each i);
2. there exists jy such that Δ( yi) ∈  for each i and ∑ i=1m Δ( yi) = 0 mod m (or Δ( yi)=∞ for each i); and
In this note we show gzs( m, 2)=5 m−4 for m≥2, gzs( m, 3)=7 m+  −6 for m≥4, gzs( m, 4)=10 m−9 for m≥3, and gzs( m, 5)=13 m−2 for m≥2.
Supported by NSF grant DMS 0097317
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