A Five Color Zero-Sum Generalization |
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Authors: | David Grynkiewicz Andrew Schultz |
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Institution: | (1) Department of Mathematics, Caltech, Pasadena, CA, 91125;(2) Department of Mathematics, Stanford University, Stanford, CA, 94305 |
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Abstract: | Let gzs(m, 2k) (gzs(m, 2k+1)) be the minimal integer such that for any coloring Δ of the integers from 1, . . . , gzs(m, 2k) by (the integers from 1 to gzs(m, 2k+1) by ) there exist integers
such that
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
1. 2(xm−x1)≤ym−x1.
In this note we show gzs(m, 2)=5m−4 for m≥2, gzs(m, 3)=7m+ −6 for m≥4, gzs(m, 4)=10m−9 for m≥3, and gzs(m, 5)=13m−2 for m≥2.
Supported by NSF grant DMS 0097317 |
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Keywords: | |
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