##### Department of Mathematics,

University of California San Diego

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### Math 269 - Combinatorics

## Dr. Marcelo Sales

#### UC Irvine

## On Pisier type problems

##### Abstract:

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.

### February 13, 2024

### 2:00 PM

APM 7321

Research Areas

Combinatorics****************************