TY - JOUR

T1 - Non-averaging Subsets and Non-vanishing Transversals

AU - Alon, Noga

AU - Ruzsa, Imre Z.

N1 - Funding Information:
* Research supported in part by a State of New Jersey grant, by a United States Israeli BSF grant, and by the Hermann Minkowski Minerva Center for Geometry at Tel Aviv University. -Research supported in part by the Hungarian National Foundation for Scientific Research, Grant Nos. 17433 and 25617.

PY - 1999/4

Y1 - 1999/4

N2 - It is shown that every set ofnintegers contains a subset of sizeΩ(n1/6) in which no element is the average of two or more others. This improves a result of Abbott. It is also proved that for everyε>0 and everym>m(ε) the following holds. IfA1,...,Amaremsubsets of cardinality at leastm1+εeach, then there area1∈A1,...,am∈Amso that the sum of every nonempty subset of the set {a1,...,am} is nonzero. This is nearly tight. The proofs of both theorems are similar and combine simple probabilistic methods with combinatorial and number theoretic tools.

AB - It is shown that every set ofnintegers contains a subset of sizeΩ(n1/6) in which no element is the average of two or more others. This improves a result of Abbott. It is also proved that for everyε>0 and everym>m(ε) the following holds. IfA1,...,Amaremsubsets of cardinality at leastm1+εeach, then there area1∈A1,...,am∈Amso that the sum of every nonempty subset of the set {a1,...,am} is nonzero. This is nearly tight. The proofs of both theorems are similar and combine simple probabilistic methods with combinatorial and number theoretic tools.

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U2 - 10.1006/jcta.1998.2926

DO - 10.1006/jcta.1998.2926

M3 - Article

AN - SCOPUS:0009917149

SN - 0097-3165

VL - 86

SP - 1

EP - 13

JO - Journal of Combinatorial Theory. Series A

JF - Journal of Combinatorial Theory. Series A

IS - 1

ER -