Non-averaging Subsets and Non-vanishing Transversals

It is shown that every set of n integers contains a subset of size Ω(n) in which no element is the average of two or more others. This improves a result of Abbott. It is also proved that for every > 0 and every m > m( ) the following holds. If A1, . . . , Am are m subsets of cardinality at least m each, then there are a1 ∈ A1, . . . , am ∈ Am so 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.