Marcelo Sales, On Pisier type problems

A subset $A\subseteq \mathbb{Z}$ of integers is free if for every two distinct subsets $B,B^{\prime }\subseteq A$ we have $$\sum _{b\in B}b\ne \sum _{b^{\prime }\in B^{\prime }}{b}^{\prime }.$$Pisier asked if for every subset $A\subseteq \mathbb{Z}$ of integers the following two statement are equivalent:

(i) $A$ is a union of finitely many free sets.
(ii) There exists $ϵ>0$ such that every finite subset $B\subseteq A$ contains a free subset $C\subseteq B$ with $|C|\ge ϵ|B|$.

In a more general framework, the Pisier question can be seen as the problem of determining if statements (i) and (ii) are equivalent for subsets of a given structure with prescribed property. We study the problem for several structures including $B_h$-sets, arithmetic progressions, independent sets in hypergraphs and configurations in the euclidean space. This is joint work with Jaroslav Nešetřil and Vojtech Rödl.