Results 1  10
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13
Zerosum sets of prescribed size
 Combinatorics, Paul Erdős is eighty
, 1993
"... Erdős, Ginzburg and Ziv proved that any sequence of 2n−1 integers contains a subsequence of cardinality n the sum of whose elements is divisible by n. We present several proofs of this result, illustrating various combinatorial and algebraic tools that have numerous other applications in Combinatori ..."
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Cited by 16 (4 self)
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Erdős, Ginzburg and Ziv proved that any sequence of 2n−1 integers contains a subsequence of cardinality n the sum of whose elements is divisible by n. We present several proofs of this result, illustrating various combinatorial and algebraic tools that have numerous other applications in Combinatorial Number Theory. Our main new results deal with an analogous multi dimensional question. We show that any sequence of 6n − 5 elements of Zn ⊕ Zn contains an nsubset the sum of whose elements is the zero vector and consider briefly the higher dimensional case as well. 1
On three zerosum Ramseytype problems
"... For a graph G whose number of edges is divisible by k, let R(G, Zk) denote the minimum integer r such that for every function f: E(Kr) ↦ → Zk there is a copy G ′ of G in Kr so that e∈E(G ′ ) f(e) = 0 (in Zk). We prove that for every integer k, R(Kn, Zk) ≤ n + O(k3 log k) provided n is sufficiently ..."
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For a graph G whose number of edges is divisible by k, let R(G, Zk) denote the minimum integer r such that for every function f: E(Kr) ↦ → Zk there is a copy G ′ of G in Kr so that e∈E(G ′ ) f(e) = 0 (in Zk). We prove that for every integer k, R(Kn, Zk) ≤ n + O(k3 log k) provided n is sufficiently large as a function of k and k divides � n 2 �. If, in addition, k is an odd primepower then R(Kn, Zk) ≤ n + 2k − 2 and this is tight if k is a prime that divides n. A related result is obtained for hypergraphs. It is further shown that for every graph G on n vertices with an even number of edges R(G, Z2) ≤ n + 2. This estimate is sharp.
Recent progress in graph pebbling
 Graph Theory Notes N. Y
"... The subject of graph pebbling has seen dramatic growth recently, both in the number of publications and in the breadth of variations and applications. Here we update the reader on the many developments that have occurred since the original Survey of Graph Pebbling in 1999. 2 1 ..."
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Cited by 2 (0 self)
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The subject of graph pebbling has seen dramatic growth recently, both in the number of publications and in the breadth of variations and applications. Here we update the reader on the many developments that have occurred since the original Survey of Graph Pebbling in 1999. 2 1
Invariant polynomials and minimal zero sequences. Under Review
"... A connection is developed between polynomials invariant under abelian permutation of their variables and minimal zero sequences in a finite abelian group. This connection is exploited to count the number of minimal invariant polynomials for various abelian groups. 1. ..."
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A connection is developed between polynomials invariant under abelian permutation of their variables and minimal zero sequences in a finite abelian group. This connection is exploited to count the number of minimal invariant polynomials for various abelian groups. 1.
ON ZEROSUM SEQUENCES IN Z/nZ ⊕ Z/nZ
"... It is well known that the maximal possible length of a minimal zerosum sequence S in the group Z/nZ⊕Z/nZ equals 2n−1, and we investigate the structure of such sequences. We say that some integer n ≥ 2 has Property B, if every minimal zerosum sequence S in Z/nZ ⊕ Z/nZ with length 2n − 1 contains so ..."
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It is well known that the maximal possible length of a minimal zerosum sequence S in the group Z/nZ⊕Z/nZ equals 2n−1, and we investigate the structure of such sequences. We say that some integer n ≥ 2 has Property B, if every minimal zerosum sequence S in Z/nZ ⊕ Z/nZ with length 2n − 1 contains some element with multiplicity n − 1. If some n ≥ 2 has Property B, then the structure of such sequences is completely determined. We conjecture that every n ≥ 2 has Property B, and we compare Property B with several other, already wellstudied properties of zerosum sequences in Z/nZ ⊕ Z/nZ. Among others, we show that if some integer n ≥ 6 has Property B, then 2n has Property B. 1.
Pseudoprimes: A Survey Of Recent Results
, 1992
"... this paper, we aim at presenting the most recent results achieved in the theory of pseudoprime numbers. First of all, we make a list of all pseudoprime varieties existing so far. This includes Lucaspseudoprimes and the generalization to sequences generated by integer polynomials modulo N , elliptic ..."
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this paper, we aim at presenting the most recent results achieved in the theory of pseudoprime numbers. First of all, we make a list of all pseudoprime varieties existing so far. This includes Lucaspseudoprimes and the generalization to sequences generated by integer polynomials modulo N , elliptic pseudoprimes. We discuss the making of tables and the consequences on the design of very fast primality algorithms for small numbers. Then, we describe the recent work of Alford, Granville and Pomerance, in which they prove that there
INVERSE ZEROSUM PROBLEMS II
, 801
"... Abstract. Let G denote a finite abelian group. The Davenport constant D(G) is the smallest integer such that each sequence over G of length at least D(G) has a nonempty zerosum subsequence, i.e., the sum of the terms equals 0 ∈ G. The constants s(G) and η(G) are defined similarly; the additional c ..."
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Abstract. Let G denote a finite abelian group. The Davenport constant D(G) is the smallest integer such that each sequence over G of length at least D(G) has a nonempty zerosum subsequence, i.e., the sum of the terms equals 0 ∈ G. The constants s(G) and η(G) are defined similarly; the additional condition that the length of the zerosum subsequence is equal to (not greater than, resp.) the exponent of G is imposed. In this paper the inverse problems associated to these constants are investigated for groups of rank two. Assuming wellsupported conjectures on this problem for groups that are the direct sum of two cyclic groups of the same order, the problems are solved for general groups of rank two. In combination with partial results towards these conjectures, this result yields unconditional results for certain types of groups of rank two. 1.
An Application of Graph Pebbling . . .
, 2008
"... A sequence of elements of a finite group G is called a zerosum sequence if it sums to the identity of G. The study of zerosum sequences has a long history with many important applications in number theory and group theory. In 1989 Kleitman and Lemke, and independently Chung, proved a strengthening ..."
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A sequence of elements of a finite group G is called a zerosum sequence if it sums to the identity of G. The study of zerosum sequences has a long history with many important applications in number theory and group theory. In 1989 Kleitman and Lemke, and independently Chung, proved a strengthening of a number theoretic conjecture of Erdős and Lemke. Kleitman and Lemke then made more general conjectures for finite groups, strengthening the requirements of zerosum sequences. In this paper we prove their conjecture in the case of abelian groups. Namely, we use graph pebbling to prove that for every sequence (gk) G k=1 of G  elements of a finite abelian group G there is a nonempty subsequence (gk)k∈K such that ∑ k∈K gk = 0G and ∑ k∈K 1/gk  ≤ 1, where g  is the order of the element g ∈ G.
REMARKS ON A GENERALIZATION OF THE DAVENPORT
, 2009
"... A generalization of the Davenport constant is investigated. For a finite abelian group G and a positive integer k, let Dk(G) denote the smallest ℓ such that each sequence over G of length at least ℓ has k disjoint nonempty zerosum subsequences. For general G, expanding on known results, upper an ..."
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A generalization of the Davenport constant is investigated. For a finite abelian group G and a positive integer k, let Dk(G) denote the smallest ℓ such that each sequence over G of length at least ℓ has k disjoint nonempty zerosum subsequences. For general G, expanding on known results, upper and lower bounds on these invariants are investigated and it is proved that the sequence (Dk(G))k∈N is eventually an arithmetic progression with difference exp(G), and several questions arising from this fact are investigated. For elementary 2groups, Dk(G) is investigated in detail; in particular, the exact values are determined for groups of rank four and five (for rank at most three they were already known).