Non-averaging subsets and non-vanishing transversals

It is shown that every set of it integers contains a subset of size Omega(n(1/6)) in which no element is the average of two or more others. This improves a result of Abbott. it is also proved that For every epsilon > 0 and every m > m(epsilon) the following holds, if A(1), ...., A(m) are m subsets of cardinality at least m(1+epsilon) each, then there are a(1) epsilon A(1), .... a(m) epsilon A(m) so that the sum of every nonempty subset of the set {a(1), .... a(m)} is nonzero. This is nearly tight. The proofs of both theorems are similar and combine simple probabilistic methods with combinatorial and number theoretic tools. (C) 1999 Academic Press.