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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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Cited by 6 (4 self)
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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.
J. Aust. Math. Soc. 94 (2013), 268–275 doi:10.1017/S1446788712000547 CARMICHAEL NUMBERS IN ARITHMETIC PROGRESSIONS
, 2013
"... We prove that when (a, m) = 1 and a is a quadratic residue mod m, there are infinitely many Carmichael numbers in the arithmetic progression a mod m. Indeed the number of them up to x is at least x 1/5 when x is large enough (depending on m). ..."
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We prove that when (a, m) = 1 and a is a quadratic residue mod m, there are infinitely many Carmichael numbers in the arithmetic progression a mod m. Indeed the number of them up to x is at least x 1/5 when x is large enough (depending on m).