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\...
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| Date: | 2016 |
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Lugansk National Taras Shevchenko University
2016
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oai:ojs.admjournal.luguniv.edu.ua: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 |
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Algebra and Discrete Mathematics |
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| datestamp_date |
2016-11-15T13:03:03Z |
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OJS |
| language |
English |
| topic |
distance sets colouring integers sets and sequences 05D10 |
| spellingShingle |
distance sets colouring integers sets and sequences 05D10 Ahmed, Tanbir On colouring integers avoiding \(t\)-AP distance-sets |
| topic_facet |
distance sets colouring integers sets and sequences 05D10 |
| format |
Article |
| author |
Ahmed, Tanbir |
| author_facet |
Ahmed, Tanbir |
| author_sort |
Ahmed, Tanbir |
| title |
On colouring integers avoiding \(t\)-AP distance-sets |
| title_short |
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_sort |
on colouring integers avoiding \(t\)-ap distance-sets |
| 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. |
| publisher |
Lugansk National Taras Shevchenko University |
| publishDate |
2016 |
| url |
https://admjournal.luguniv.edu.ua/index.php/adm/article/view/78 |
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AT ahmedtanbir oncolouringintegersavoidingtapdistancesets |
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2025-07-17T10:33:54Z |
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2025-07-17T10:33:54Z |
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