a subset $a\subseteq\mathbf{z}$ of integers is free if for every two distinct subsets $b,b'\subseteq a$ we have $$\sum_{b\in b}b\neq\sum_{b'\in b'}b'.$$ pisier asked if for every subset $a\subseteq\mathbf{z}$ of integers the following two statement are equivalent:
(i) $a$ is a union of finitely many free sets.
(ii) there exists $\varepsilon>0$ such that every finite subset $b\subseteq a$ contains a free subset $c\subseteq b$ with $\vert c\vert\geq \varepsilon \vert b\vert$.
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, christian reiher and vojtěch rödl.