A Five Color Zero-Sum Generalization

AbstractLet 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 that1. 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); and1. 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.

[1]  David J. Grynkiewicz,et al.  On Four Colored Sets with Nondecreasing Diameter and the Erds-Ginzburg-Ziv Theorem , 2002, J. Comb. Theory, Ser. A.

[2]  John E. Olson.  An addition theorem for finite Abelian groups , 1977 .

[3]  Daniel Schaal,et al.  A zero-sum theorem , 2003, J. Comb. Theory, Ser. A.

[4]  Paul Baginski.  A generalization of a Ramsey-type theorem on hypermatchings , 2005 .

[5]  A. Ziv,et al.  Theorem in the Additive Number Theory , 2022 .

[6]  Arie Bialostocki,et al.  On constrained 2-Partitions of monochromatic sets and generalizations in the sense of Erdos-Ginzburg-Ziv , 2005, Ars Comb..

[7]  Yair Caro,et al.  Zero-sum problems - A survey , 1996, Discret. Math..

[8]  David J. Grynkiewicz,et al.  On four color monochromatic sets with nondecreasing diameter , 2005, Discret. Math..

[9]  Daniel Schaal,et al.  On a Zero-Sum Generalization of a Variation of Schur’s Equation , 2008, Graphs Comb..

[10]  Zoltán Füredi,et al.  On zero-trees , 1992, J. Graph Theory.

[11]  Paul D. Seymour,et al.  A simpler proof and a generalization of the zero-trees theorem , 1991, J. Comb. Theory, Ser. A.

[12]  Arie Bialostocki,et al.  On zero sum Ramsey numbers: Multiple copies of a graph , 1994, J. Graph Theory.

[13]  D. Grynkiewicz,et al.  Sumsets, Zero-Sums and Extremal Combinatorics , 2006 .

[14]  Andrew Schultz.  On a modification of a problem of Bialostocki, Erdos, and Lefmann , 2006, Discret. Math..

[15]  Carl R. Yerger.  Monochromatic and Zero-Sum Sets of Nondecreasing Diameter , 2005 .

[16]  Paul Baginski.  A generalization of a Ramsey-type theorem on hypermatchings , 2005, J. Graph Theory.

[17]  David J. Grynkiewicz,et al.  On a partition analog of the Cauchy-Davenport theorem , 2005 .

[18]  Roger Crocker,et al.  A theorem in additive number theory , 1969 .