On Pisier Type Theorems

For any integer h 2 h ⩾ 2 , a set of integers B=ᵢ∈ I B = { b i } i ∈ I is a Bₕ B h -set if all h -sums bi₁+… +biₕ b i 1 + … + b i h with i₁<… <iₕ i 1 < … < i h are distinct. Answering a question of Alon and Erdős [2], for every h 2 h ⩾ 2 we construct a set of integers X which is not a union of finitely many Bₕ B h -sets, yet any finite subset Y⊆ X Y ⊆ X contains an Bₕ B h -set Z with |Z| ε |Y| | Z | ⩾ ε | Y | , where ε :=ε (h) 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 Bₕ B h -sets by the requirement that all finite sums ∑ j∈ Jbⱼ ∑ j ∈ J b j are distinct.