Results 1  10
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23
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
ZEROSUM PROBLEMS IN ABELIAN pGROUPS AND COVERS OF THE INTEGERS BY RESIDUE CLASSES
 ACCEPTED BY ISRAEL J. MATH.
, 2007
"... Zerosum problems in abelian groups and covers of the integers by residue classes, are two different active topics initiated by P. Erdős more than 40 years ago and investigated by many researchers separately since then. In an earlier announcement [Electron. Res. Announc. Amer. Math. Soc. 9(2003), 51 ..."
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Cited by 9 (6 self)
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Zerosum problems in abelian groups and covers of the integers by residue classes, are two different active topics initiated by P. Erdős more than 40 years ago and investigated by many researchers separately since then. In an earlier announcement [Electron. Res. Announc. Amer. Math. Soc. 9(2003), 5160], the author claimed some surprising connections among these seemingly unrelated fascinating areas. In this paper we establish further connections between zerosum problems in abelian pgroups and covers of the integers. For example, we extend the famous ErdősGinzburgZiv theorem in the following way: If {as(mod ns)} k s=1 covers each integer either exactly 2q −1 times or exactly 2q times where q is a prime power, then for any c1,..., ck ∈ Z/qZ there exists an I ⊆ {1,..., k} such that P s∈I 1/ns = q and P s∈I cs = 0. Our main theorem in this paper unifies many results in the two realms and also implies an extension of the AlonFriedlandKalai result on regular subgraphs.
Noncanonical extensions of ErdősGinzburgZiv theorem
 Integers
, 2002
"... In 1961, ErdősGinzburgZiv proved that for a given natural number n ≥ 1 and a sequence a1,a2, ···,a2n−1 of integers (not necessarily distinct), there exist 1 ≤ i1
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Cited by 7 (1 self)
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In 1961, ErdősGinzburgZiv proved that for a given natural number n ≥ 1 and a sequence a1,a2, ···,a2n−1 of integers (not necessarily distinct), there exist 1 ≤ i1 <i2 < ·· · < in ≤ 2n −1 such that ai1 +ai2 + ···+ain is divisible by n. Moreover, the constant 2n −1 is tight. By now, there are many canonical generalizations of this theorem. In this paper, we shall prove some noncanonical generalizations of this theorem.
LocationCorrecting Codes
 IEEE Trans. Inform. Theory
, 1997
"... We study codes over GF (q) that can correct t channel errors assuming the error values are known. This is a counterpart to the wellknown problem of erasure correction, where error values are found assuming the locations are known. The correction capabilities of these so called tlocation correcting ..."
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Cited by 7 (1 self)
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We study codes over GF (q) that can correct t channel errors assuming the error values are known. This is a counterpart to the wellknown problem of erasure correction, where error values are found assuming the locations are known. The correction capabilities of these so called tlocation correcting codes (tLCCs) are characterized by a new metric, the decomposability distance, which plays a role analogous to that of the Hamming metric in conventional errorcorrecting codes (ECCs). Based on the new metric, we present bounds on the parameters of t LCCs that are counterparts to the classical Singleton, sphere packing and GilbertVarshamov bounds for ECCs. In particular, we show examples of perfect LCCs, and we study optimal (MDSlike) LCCs that attain the Singletontype bound on the redundancy. We show that these optimal codes are generally much shorter than their erasure (or conventional ECC) analogues: The length n of any tLCC that attains the Singletontype bound for t ? 1 is bounde...
CASTELNUOVOMUMFORD REGULARITY BY APPROXIMATION
, 2003
"... Abstract. The CastelnuovoMumford regularity of a module gives a rough measure of its complexity. We bound the regularity of a module given a system of approximating modules whose regularities are known. Such approximations can arise naturally for modules constructed by inductive combinatorial means ..."
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Cited by 5 (2 self)
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Abstract. The CastelnuovoMumford regularity of a module gives a rough measure of its complexity. We bound the regularity of a module given a system of approximating modules whose regularities are known. Such approximations can arise naturally for modules constructed by inductive combinatorial means. We apply these methods to bound the regularity of ideals constructed as combinations of linear ideals and the module of derivations of a hyperplane arrangement as well as to give degree bounds for invariants of finite groups. 1.
A UNIFIED THEORY OF ZEROSUM PROBLEMS, SUBSET SUMS AND COVERS OF Z
, 2004
"... Zerosum problems on abelian groups, subset sums in a field and covers of the integers by residue classes, are three different active topics initiated by P. Erdős and investigated by many researchers. In an earlier ..."
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Cited by 3 (3 self)
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Zerosum problems on abelian groups, subset sums in a field and covers of the integers by residue classes, are three different active topics initiated by P. Erdős and investigated by many researchers. In an earlier
On zerosum subsequences in finite abelian groups
 Integers: Electronic Journal Comb. Number Th
"... Let G be a finite abelian group and k ∈ N with k ∤ exp(G). Then Ek(G) denotes the smallest integer l ∈ N such that every sequence S ∈F(G) with S  ≥lhas a zerosum subsequence T with k ∤ T . In this paper we prove that if G = Cn1 ⊕···⊕Cnr is a pgroup, k ∈ N with k ∤ exp(G) and gcd(p, k) = 1, th ..."
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Cited by 2 (0 self)
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Let G be a finite abelian group and k ∈ N with k ∤ exp(G). Then Ek(G) denotes the smallest integer l ∈ N such that every sequence S ∈F(G) with S  ≥lhas a zerosum subsequence T with k ∤ T . In this paper we prove that if G = Cn1 ⊕···⊕Cnr is a pgroup, k ∈ N with k ∤ exp(G) and gcd(p, k) = 1, then r�
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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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
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.