Almost disjoint families of 3-term arithmetic progressions
type: Item
instance of: scholarly article
title: Almost disjoint families of 3-term arithmetic progressions
zbMATH Open document ID: 1064.05145
author: Hayri Ardal
author: Peter A. B. Pleasants
author: Tom Brown
published in: Journal of Combinatorial Theory. Series A
publication date: 2005-02-22T00:00:00Z
review text: In the paper the problem of the maximum size of a family of \(3\)-term arithmetic progressions (APs) of integers from the interval \([0,n-1]\) with the property that not two of them have more than one term in common (called almost disjoint in the paper) is investigated. The total number of \(3\)-term APs in \([0,n-1]\) is \(n^2/4+O(n)\). The main result of the paper shows that the maximum size of a family of almost disjoint \(3\)--term APs in \([0,n-1]\) is asymptotically equal to \(C(n^2/4)+ O(n^{2/3}\log n)\) for some \(C\in(0.4676,0.485)\). For the number \(H(n)\) of maximum-sized families of almost disjoint \(3\)-term APs in \([0,n-1]\) the authors prove the asymptotic relation \(\log_2 H(n)=E(n^2/4)+O(n^{5/3}\log n)\) with \(E\in(0.102,0.447)\). Based on the observation that two \(3\)-term APs with two numbers in common either have the same common difference or the common difference of one is exactly twice that of the other, the authors also investigate related problems for families the \(3\)-term APs whose common difference is a power of \(2\).
reviewed by: Štefan Porubský
Mathematics Subject Classification ID: 05D99
Mathematics Subject Classification ID: 11B25
zbMATH DE Number: 2136713
MaRDI profile type: Publication
full work available at URL: j.jcta.2004.07.010
OpenAlex ID: W2022449958
cites work: Intersection Properties of Systems of Finite Sets
cites work: Q4871772
cites work: On a packing and covering problem
DOI: 10.1016/J.JCTA.2004.07.010
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description: scientific article; zbMATH DE number 2136713