# On Pisier type problems Open Access

## Sales, Marcelo (Spring 2023)

Permanent URL: https://etd.library.emory.edu/concern/etds/5t34sk889?locale=en
Published

## Abstract

A subset $A\subseteq \ZZ$ of integers is \textit{free} if for every two distinct subsets $B, B'\subseteq A$ we have

\begin{align*}

\sum_{b\in B}b\neq \sum_{b'\in B'} b'.

\end{align*}

Pisier asked if for every subset $A\subseteq \ZZ$ of integers the following two statement are equivalent:

\begin{enumerate}

\item[(i)] $A$ is a union of finitely many free sets.

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

\end{enumerate}

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.

1

Introduction

1

1.1

Arithmetic progressions and

B

h

-sets . . . . . . . . . . . . . . . . . . .

3

1.2

Euclidean configurations

. . . . . . . . . . . . . . . . . . . . . . . . .

6

1.3

Organization

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

2

Independent sets on hypergraphs

10

2.1

μ

-fractional property

. . . . . . . . . . . . . . . . . . . . . . . . . . .

11

2.2

A version for simple graphs

. . . . . . . . . . . . . . . . . . . . . . .

15

2.3

Independent sets of shift graphs . . . . . . . . . . . . . . . . . . . . .

18

3

Pisier type problem for

B

h

-sets

20

3.1

A local version of the Pisier problems for sets

. . . . . . . . . . . . .

20

3.2

Proof of Theorem 1.1.5 . . . . . . . . . . . . . . . . . . . . . . . . . .

30

4

Pisier type problems for arithmetic progressions

34

4.1

A modification of Hales–Jewett theorem

. . . . . . . . . . . . . . . .

34

4.2

The partite construction

. . . . . . . . . . . . . . . . . . . . . . . . .

36

4.3

A property of the construction . . . . . . . . . . . . . . . . . . . . . .

42

4.4

Proof of Theorem 4.2.1 . . . . . . . . . . . . . . . . . . . . . . . . . .

49

5

Euclidean configurations

52

5.1

Segments are P-Ramsey

. . . . . . . . . . . . . . . . . . . . . . . . .

52

5.2

Robust configurations . . . . . . . . . . . . . . . . . . . . . . . . . . .

54

5.3

Simplices are P-Ramsey

. . . . . . . . . . . . . . . . . . . . . . . . .

59

6

Concluding remarks

66

6.1

Pisier type problems for linear system of equations

. . . . . . . . . .

66

6.2

Euclidean considerations

. . . . . . . . . . . . . . . . . . . . . . . . .

67

Bibliography

69