Discrete Analysis Seminar

A subset $A\subseteq\mathds{Z}$ 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 $A\subseteq\mathds{Z}$ 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 and Vojtech Rödl.