2003
DOI: 10.1017/s0963548303005716
|View full text |Cite
|
Sign up to set email alerts
|

Open Problems in Partition Regularity

Abstract: A finite or infinite matrix A with rational entries is called partition regular if, whenever the natural numbers are finitely coloured, there is a monochromatic vector x with . Many of the classical theorems of Ramsey Theory may naturally be interpreted as assertions that particular matrices are partition regular.While in the finite case partition regularity is well understood, very little is known in the infinite case. Our aim in this paper is to present some of the natural and appealing open problems in the … Show more

Help me understand this report

Search citation statements

Order By: Relevance

Paper Sections

Select...
4
1

Citation Types

0
53
0
1

Year Published

2006
2006
2021
2021

Publication Types

Select...
5
3

Relationship

1
7

Authors

Journals

citations
Cited by 51 publications
(54 citation statements)
references
References 27 publications
0
53
0
1
Order By: Relevance
“…Rado made a beautiful conjecture in his thesis that further differentiates those systems of linear equations that are regular from those that are not regular. This outstanding conjecture, known as Rado's Boundedness Conjecture, has remained open for all but the trivial cases [17].…”
Section: Introductionmentioning
confidence: 99%
See 1 more Smart Citation
“…Rado made a beautiful conjecture in his thesis that further differentiates those systems of linear equations that are regular from those that are not regular. This outstanding conjecture, known as Rado's Boundedness Conjecture, has remained open for all but the trivial cases [17].…”
Section: Introductionmentioning
confidence: 99%
“…Over the past seven decades, Rado's Boundedness Conjecture has received considerable attention [3][4][5]8,[15][16][17]23]. Deuber [8] called the problem "intriguing", while more recently, Hindman, Leader, and Strauss [17] called it one of the major open questions in partition regularity.…”
Section: Introductionmentioning
confidence: 99%
“…Например, до сих пор нет ответа на во-прос, существует ли монохроматическое решение у уравнения x 2 + y 2 = z 2 , а также на вопрос, для любой ли конечной раскраски Z найдутся x + y, xy одного и того же цвета (см. [109], проблема 3). Неизвестно решение и более сложного вопроса: при всякой ли раскраске существуют x, y, x + y, xy, име-ющие один и тот же цвет?…”
Section: задачи о раскраскахunclassified
“…One example is the Finite Sums Theorem [5], which states that whenever the positive integers are finitely coloured, there exists a sequence x n ∞ n=0 such that { n∈F x n : F is a finite nonempty subset of ω} is monochromatic. For other examples see [8] and [1].…”
Section: Introductionmentioning
confidence: 99%
“…(And, as noted in [8], the corresponding statement applies to infinite kernel partition regular matrices. This fact also follows from Theorem 2.8 below.…”
Section: Introductionmentioning
confidence: 99%