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# Marcelo Sales, On Pisier type problems

## Thursday, March 9, 2023 @ 10:00 AM - 11:00 AM KST

A subset $A\subseteq \mathbb Z$ 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 $A\subseteq \mathbb Z$ of integers the following two statement are equivalent:

(i) $A$ is a union of finitely many free sets.

(ii) There exists $\epsilon>0$ such that every finite subset $B\subseteq A$ contains a free subset $C\subseteq B$ with $|C|\geq \epsilon |B|$.

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.