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Discrete Analysis Seminar

Tuesday, April 25, 2023
3:00pm to 4:00pm
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Linde Hall 187
On Pisier type problems
Marcelo Sales, Department of Mathematics, Emory University,

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.

For more information, please contact Math Department by phone at 626-395-4335 or by email at [email protected].