On Four Colored Sets with Nondecreasing Diameter and the Erdős–Ginzburg–Ziv Theorem

Grynkiewicz, David J.

doi:10.1006/jcta.2002.3277

Cited by 16 publications

(8 citation statements)

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“…Any Ramsey theory problem has a corresponding zero-sum version obtained by replacing colorings using two colors with colorings using the elements of Z m and looking for zero-sum substructures rather than monochromatic ones. Through a slightly more complicated way, Ramsey theory questions involving more than two colors also have a corresponding zero-sum version [7], [34], [35]. If m is chosen to be the size of the particular substructure in question, then the zero-sum Ramsey number always gives an upper bound on the monochromatic Ramsey number, but, perhaps unexpectedly, the two numbers were in many cases equal.…”

Section: Introduction and Main Resultsmentioning

confidence: 97%

“…The following references also contain several theorems that are often quite useful when used in conjunction with Corollary 1. Corollary 1 was used to establish a four-color zero-sum generalization that was the next open case in a conjecture of Bialostocki, Erdős, and Lefmann [7], [35]. Relaxing the structure from the aforementioned problem, a zero-sum generalization was obtained, with the aid of the Olson case of Corollary 1, for two, three, four and five colors [34].…”

Section: Introduction and Main Resultsmentioning

confidence: 98%

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On a partition analog of the Cauchy-Davenport theorem

Grynkiewicz

2005

Acta Math Hung

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Let G be a finite abelian group, and let n be a positive integer. From the Cauchy-Davenport Theorem it follows that if G is a cyclic group of prime order, then any collection of n subsets A1, A2, . . . , An of G satisfiesM. Kneser generalized the Cauchy-Davenport Theorem for any abelian group. In this paper, we prove a sequence-partition analog of the Cauchy-Davenport Theorem along the lines of Kneser's Theorem. A particular case of our theorem was proved by J. E. Olson in the context of the Erdős-Ginzburg-Ziv Theorem.

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Section: Introduction and Main Resultsmentioning

confidence: 97%

Section: Introduction and Main Resultsmentioning

confidence: 98%

On a partition analog of the Cauchy-Davenport theorem

Grynkiewicz

2005

Acta Math Hung

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“…, m; r) in [3], where they determined that f (m, m; 2) = 5m − 3, that f (m, m; 3) = 9m − 7 and that (1) 8m − 4 ≤ f (m, m, m; 2) ≤ 10m − 6, as well as giving asymptotic bounds for t = 2. The problem was motivated in part by zero-sum generalizations in the sense of the Erdős-Ginzburg-Ziv Theorem [6] [9, Theorem 10.1] (see [2] [3] [7] for a short discussion of zero-sum generalizations, including definitions). Subsequently, Bollobás, Erdős, and Jin obtained improved results for m = 2, showing that 4r − log 2 r + 1 ≤ f * (2, 2; r) ≤ 4r + 1 and f * (2, 2; 2 k ) = 4 · 2 k + 1, as well as giving improved asymptotic bounds for t and r when m = 2.…”

Section: Introductionmentioning

confidence: 99%

On three sets with nondecreasing diameter

Bernstein

Grynkiewicz

Yerger

2015

Discrete Mathematics

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“…They also introduced a notion of zero-sum generalization for Ramsey-type problems involving arbitrary r-colorings (not just 2-colorings), and showed that the corresponding 3-color version of the nondecreasing diameter problem for two m-sets also zero-sum generalized. Recently, the four color case was shown to zero-sum generalize [24], but the cases with r > 4 remain open and difficult.…”

Section: Introductionmentioning

confidence: 99%

“…In this paper we introduce and study the functions f j (m, 2) and f j (m, Z m ) with j < m, thus studying the nondecreasing diameter problem by varying the notion of diameter by the parameter j. One of our main tools is an improvement to a recent generalization (Theorem 2.7) of results of Mann [29], Olson [31], Bollobás and Leader [10], and Hamidoune [26], that was developed by the first author [23] while studying the original nondecreasing diameter problem for four colors [24].…”

Section: Introductionmentioning

confidence: 99%

Monochromatic and Zero-Sum Sets of Nondecreasing Modified Diameter

Grynkiewicz¹,

Sabar²

2006

Electron. J. Combin.

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Let $m$ be a positive integer whose smallest prime divisor is denoted by $p$, and let ${\Bbb Z}_m$ denote the cyclic group of residues modulo $m$. For a set $B=\{x_1,x_2,\ldots,x_m\}$ of $m$ integers satisfying $x_1 < x_2 < \cdots < x_m$, and an integer $j$ satisfying $2\leq j \leq m$, define $g_j(B)=x_j-x_1$. Furthermore, define $f_j(m,2)$ (define $f_j(m,{\Bbb Z}_m)$) to be the least integer $N$ such that for every coloring $\Delta : \{1,\ldots,N\}\rightarrow \{0,1\}$ (every coloring $\Delta : \{1,\ldots,N\}\rightarrow {\Bbb Z}_m$), there exist two $m$-sets $B_1,B_2\subset \{1,\ldots,N\}$ satisfying: (i) $\max(B_1) < \min(B_2)$, (ii) $g_j(B_1)\leq g_j(B_2)$, and (iii) $|\Delta (B_i)|=1$ for $i=1,2$ (and (iii) $\sum_{x\in B_i}\Delta (x)=0$ for $i=1,2$). We prove that $f_j(m,2)\leq 5m-3$ for all $j$, with equality holding for $j=m$, and that $f_j(m,{\Bbb Z}_m)\leq 8m+{m\over p}-6$. Moreover, we show that $f_j(m,2)\ge 4m-2+(j-1)k$, where $k=\left\lfloor\left(-1+\sqrt{{8m-9+j\over j-1}}\right){/2}\right\rfloor$, and, if $m$ is prime or $j\geq{m\over p}+p-1$, that $f_j(m,{\Bbb Z}_m)\leq 6m-4$. We conclude by showing $f_{m-1}(m,2)=f_{m-1}(m,{\Bbb Z}_m)$ for $m\geq 9$.

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On Four Colored Sets with Nondecreasing Diameter and the Erdős–Ginzburg–Ziv Theorem

Cited by 16 publications

References 9 publications

On a partition analog of the Cauchy-Davenport theorem

On a partition analog of the Cauchy-Davenport theorem

On three sets with nondecreasing diameter

Monochromatic and Zero-Sum Sets of Nondecreasing Modified Diameter

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