On colouring integers avoiding \(t\)-AP distance-sets
A \(t\)-AP is a sequence of the form \(a,a+d,\ldots, a+(t-1)d\),where \(a,d\in \mathbb{Z}\). Given a finite set \(X\) and positive integers \(d\), \(t\), \(a_1,a_2,\ldots,a_{t-1}\), define \(\nu(X,d) = |\{(x,y):{x,y\in{X}},{y>x},{y-x=d}\}|\), \((a_1,a_2,\ldots,a_{t-1};d) =\) a collection \(X\...
Збережено в:
| Дата: | 2016 |
|---|---|
| Автор: | |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Lugansk National Taras Shevchenko University
2016
|
| Теми: | |
| Онлайн доступ: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/78 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Algebra and Discrete Mathematics |
Репозитарії
Algebra and Discrete Mathematics| _version_ | 1856543436383977472 |
|---|---|
| author | Ahmed, Tanbir |
| author_facet | Ahmed, Tanbir |
| author_sort | Ahmed, Tanbir |
| baseUrl_str | |
| collection | OJS |
| datestamp_date | 2016-11-15T13:03:03Z |
| description | A \(t\)-AP is a sequence of the form \(a,a+d,\ldots, a+(t-1)d\),where \(a,d\in \mathbb{Z}\). Given a finite set \(X\) and positive integers \(d\), \(t\), \(a_1,a_2,\ldots,a_{t-1}\), define \(\nu(X,d) = |\{(x,y):{x,y\in{X}},{y>x},{y-x=d}\}|\), \((a_1,a_2,\ldots,a_{t-1};d) =\) a collection \(X\) s.t. \(\nu(X,d\cdot{i})\geq a_i\) for \(1\leq i\leq t-1\).In this paper, we investigatethe structure of sets with bounded number of pairs with certain gaps.Let \((t-1,t-2,\ldots,1; d)\) be called a \emph{\(t\)-AP distance-set} of size at least \(t\).A \(k\)-colouring of integers \(1,2,\ldots, n\) is a mapping \(\{1,2,\ldots,n\}\rightarrow \{0,1,\ldots,k-1\}\) where\(0,1,\ldots,k-1\) are colours.Let \(ww(k,t)\) denote thesmallest positive integer \(n\) such that every \(k\)-colouring of \(1,2,\ldots,n\)contains a monochromatic \(t\)-AP distance-set for some \(d>0\).We conjecture that \(ww(2,t)\geq t^2\) and prove the lower bound for most cases.We also generalize the notion of \(ww(k,t)\) and prove several lower bounds. |
| first_indexed | 2025-12-02T15:43:20Z |
| format | Article |
| id | admjournalluguniveduua-article-78 |
| institution | Algebra and Discrete Mathematics |
| language | English |
| last_indexed | 2025-12-02T15:43:20Z |
| publishDate | 2016 |
| publisher | Lugansk National Taras Shevchenko University |
| record_format | ojs |
| spelling | admjournalluguniveduua-article-782016-11-15T13:03:03Z On colouring integers avoiding \(t\)-AP distance-sets Ahmed, Tanbir distance sets, colouring integers, sets and sequences 05D10 A \(t\)-AP is a sequence of the form \(a,a+d,\ldots, a+(t-1)d\),where \(a,d\in \mathbb{Z}\). Given a finite set \(X\) and positive integers \(d\), \(t\), \(a_1,a_2,\ldots,a_{t-1}\), define \(\nu(X,d) = |\{(x,y):{x,y\in{X}},{y>x},{y-x=d}\}|\), \((a_1,a_2,\ldots,a_{t-1};d) =\) a collection \(X\) s.t. \(\nu(X,d\cdot{i})\geq a_i\) for \(1\leq i\leq t-1\).In this paper, we investigatethe structure of sets with bounded number of pairs with certain gaps.Let \((t-1,t-2,\ldots,1; d)\) be called a \emph{\(t\)-AP distance-set} of size at least \(t\).A \(k\)-colouring of integers \(1,2,\ldots, n\) is a mapping \(\{1,2,\ldots,n\}\rightarrow \{0,1,\ldots,k-1\}\) where\(0,1,\ldots,k-1\) are colours.Let \(ww(k,t)\) denote thesmallest positive integer \(n\) such that every \(k\)-colouring of \(1,2,\ldots,n\)contains a monochromatic \(t\)-AP distance-set for some \(d>0\).We conjecture that \(ww(2,t)\geq t^2\) and prove the lower bound for most cases.We also generalize the notion of \(ww(k,t)\) and prove several lower bounds. Lugansk National Taras Shevchenko University 2016-11-15 Article Article Peer-reviewed Article application/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/view/78 Algebra and Discrete Mathematics; Vol 22, No 1 (2016) 2415-721X 1726-3255 en https://admjournal.luguniv.edu.ua/index.php/adm/article/view/78/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/downloadSuppFile/78/16 https://admjournal.luguniv.edu.ua/index.php/adm/article/downloadSuppFile/78/40 Copyright (c) 2016 Algebra and Discrete Mathematics |
| spellingShingle | distance sets colouring integers sets and sequences 05D10 Ahmed, Tanbir On colouring integers avoiding \(t\)-AP distance-sets |
| title | On colouring integers avoiding \(t\)-AP distance-sets |
| title_full | On colouring integers avoiding \(t\)-AP distance-sets |
| title_fullStr | On colouring integers avoiding \(t\)-AP distance-sets |
| title_full_unstemmed | On colouring integers avoiding \(t\)-AP distance-sets |
| title_short | On colouring integers avoiding \(t\)-AP distance-sets |
| title_sort | on colouring integers avoiding \(t\)-ap distance-sets |
| topic | distance sets colouring integers sets and sequences 05D10 |
| topic_facet | distance sets colouring integers sets and sequences 05D10 |
| url | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/78 |
| work_keys_str_mv | AT ahmedtanbir oncolouringintegersavoidingtapdistancesets |