On Pisier Type Theorems

For any integer \(h\geqslant 2\) $h⩾2$ , a set of integers \(B=\{b_i\}_{i\in I}\) $B={\{b_i\}}_{i\in I}$ is a \(B_h\) $B_h$ -set if all h-sums \(b_{i_1}+\ldots +b_{i_h}\) $b_{i_1}+\dots +b_{i_h}$ with \(i_1<\ldots <i_h\) $i_1<\dots <i_h$ are distinct. Answering a question of Alon and Erdős [2], for every \(h\geqslant 2\) $h⩾2$ we construct a set of integers X which is not a union of finitely many \(B_h\) $B_h$ -sets, yet any finite subset \(Y\subseteq X\) $Y\subseteq X$ contains an \(B_h\) $B_h$ -set Z with \(|Z|\geqslant \varepsilon |Y|\) $|Z|⩾\epsilon |Y|$ , where \(\varepsilon :=\varepsilon (h)\) $\epsilon :=\epsilon (h)$ . We also discuss questions related to a problem of Pisier about the existence of a set A with similar properties when replacing \(B_h\) $B_h$ -sets by the requirement that all finite sums \(\sum _{j\in J}b_j\) ${\sum }_{j\in J}{b}_j$ are distinct.