On Pisier Type Theorems.

For any integer h ⩾ 2 , a set of integers B = { b i } i ∈ I is a B h -set if all h-sums b i 1 + ... + b i h with i 1 < ... < i h are distinct. Answering a question of Alon and Erdős [2], for every h ⩾ 2 we construct a set of integers X which is not a union of finitely many B h -sets, yet any finite subset Y ⊆ X contains an B h -set Z with | Z | ⩾ ε | Y | , where ε : = ε (h) . We also discuss questions related to a problem of Pisier about the existence of a set A with similar properties when replacing B h -sets by the requirement that all finite sums ∑ j ∈ J b j are distinct. [ABSTRACT FROM AUTHOR]