Non-averaging Subsets and Non-vanishing Transversals

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.