A variant of Kemnitz Conjecture

Gao, Weidong; Thangadurai, R.

doi:10.1016/j.jcta.2004.03.009

Cited by 20 publications

(17 citation statements)

References 16 publications

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“…Then the equality holds for n = 2 (trivial) and n = 3 [34,Hilfssatz 3]. If p is a prime with p ≥ 67, then g(C p ⊕ C p ) = s(C p ) = 2p − 1; see Theorem 1.1 and [29].…”

Section: Definition 21 We Denote Bymentioning

confidence: 99%

Zero-Sum Problems in Finite Abelian Groups and Affine Caps

Edel

Elsholtz

Geroldinger

et al. 2006

The Quarterly Journal of Mathematics

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For a finite abelian group G, let s(G) denote the smallest integer l such that every sequence S over G of length |S| ≥ l has a zero-sum subsequence of length exp(G). We derive new upper and lower bounds for s(G), and all our bounds are sharp for special types of groups. The results are not restricted to groups G of the form G = C r n , but they respect the structure of the group. In particular, we show s(C 4 n) ≥ 20n − 19 for all odd n, which is sharp if n is a power of 3. Moreover, we investigate the relationship between extremal sequences and maximal caps in finite geometry.

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“…Then the equality holds for n = 2 (trivial) and n = 3 [34,Hilfssatz 3]. If p is a prime with p ≥ 67, then g(C p ⊕ C p ) = s(C p ) = 2p − 1; see Theorem 1.1 and [29].…”

Section: Definition 21 We Denote Bymentioning

confidence: 99%

Zero-Sum Problems in Finite Abelian Groups and Affine Caps

Edel

Elsholtz

Geroldinger

et al. 2006

The Quarterly Journal of Mathematics

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“…When G is cyclic of order m, we have h(G, k) ≥ k + 1 provided m − ⌊m/4⌋ − 2 ≤ k ≤ m − 2 (see [8]); h(G, k) ≥ k + 1 provided m is prime with 1 ≤ k ≤ m − 2 (see [11]); h(G, m − 2) = m − 1 (see [1] or [15]); and h(G, m − 3) = m − 1 (see [7]). …”

Section: Results We Have the Following Open Problemmentioning

confidence: 99%

On Erdős–Ginzburg–Ziv inverse theorems

et al. 2007

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1. Introduction. Let F (G) denote the free abelian monoid over the set G with monoid operation written multiplicatively and given by concatenation, i.e., F (G) consists of all finite sequences over G modulo the equivalence relation allowing terms to be permuted. Despite possible confusion, the elements of F (G) will be referred to simply as sequences, and if indeed order or being infinite are needed in a sequence, it will be explicitly stated when the sequence is first introduced. Now let G be an abelian group of order m ≥ 2. The Erdős-GinzburgZiv theorem states that every sequence in G of length 2m − 1 contains an m-term subsequence with zero sum [5]. There have been many related inverse theorems describing the structure of the sequences S in G with length |S| = m + k, 1 ≤ k ≤ m − 2, not having any m-term subsequence with zero sum. For cyclic groups of order m, the structure of S has been described by several authors: when k = m − 2, by Yuster and Peterson in [15], and by Bialostocki and Dierker in [1]; when k = m − 3, by Flores and Ordaz in [7]; when m − ⌊m/4⌋ − 2 ≤ k ≤ m − 2, by Bialostocki, Dierker, Grynkiewicz, and Lotspeich in [2] (using a related result of Gao from [8]); and when k ≥ ⌈(m − 1)/2⌉, by Chen and Savchev in [3].

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“…In particular, g(C 2 3 ) = 5. More recently Gao and Thangadurai [4] showed g(C 2 p ) = 2p−1 for prime p ≥ 67 and g(C 2 4 ) = 9. In [2] we can find other values for elementary 3-group; for example g(C 3 3 ) = 10, g(C 3 3 ) = 21, g(C 5 3 ) = 46 and 112 ≤ g(C 6 3 ) ≤ 114 [1,7,8,12].…”

Section: Remark 2 Kemnitz Showed G(cmentioning

confidence: 99%

Existence conditions for k- barycentric Olson constant

Marchan

Ordaz

Villaroel

et al. 2020

RMTA

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Let (G, +) be a finite abelian group and 3 ≤ k ≤ |G| a positive integer. The k-barycentric Olson constant denoted by BO(k, G) is defined as the smallest integer ℓ such that each set A of G with |A| = ℓ contains a subset with k elements {a1, . . . , ak} satisfying a1 + · · · + ak = kaj for some 1 ≤ j ≤ k. We establish some general conditions on G assuring the existence of BO(k, G) for each 3 ≤ k ≤ |G|. In particular, from our results we can derive the existence conditions for cyclic groups and for elementary p-groups p ≥ 3. We give a special treatment over the existence condition for the elementary 2-groups.

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A variant of Kemnitz Conjecture

Cited by 20 publications

References 16 publications

Zero-Sum Problems in Finite Abelian Groups and Affine Caps

Zero-Sum Problems in Finite Abelian Groups and Affine Caps

On Erdős–Ginzburg–Ziv inverse theorems

Existence conditions for k- barycentric Olson constant

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