On Pisier type problems Público
Sales, Marcelo (Spring 2023)
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.
Table of Contents
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
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