Maximal collections of intersecting arithmetic progressions
type: Item
instance of: scholarly article
title: Maximal collections of intersecting arithmetic progressions
zbMATH Open document ID: 1046.05077
author: Kevin Ford
published in: Combinatorica
publication date: 2004-09-07T00:00:00Z
review text: Let \(N_t(k)\), \(t\geq2\), be the maximum number of \(k\)-term arithmetic progressions of real numbers, any couple of which have \(t\) common elements. In the paper it is proved that \(N_2(t)=k(k-1)/2\) if \(k\) is a prime. Then the author proves that the same conclusion is true for \(k\geq10^{8000}\). The technique of proof (employing also a strong form of the theorem of \textit{R. Balasubramanian} and \textit{K. Soundararajan} [Acta Arith. 75, 1-38 (1996; Zbl 0853.11002)]) does not allow to lower this bound due to the lack of better bounds for primes \(n\) short intervals. For the case \(t\geq 3\) less precise lower and upper bounds for \(N_t(k)\) are proved.
Mathematics Subject Classification ID: 05D05
Mathematics Subject Classification ID: 11B75
Mathematics Subject Classification ID: 11B25
Mathematics Subject Classification ID: 11A05
Mathematics Subject Classification ID: 11N05
Mathematics Subject Classification ID: 11N37
zbMATH DE Number: 2099075
zbMATH Keywords: arithmetic progression
zbMATH Keywords: Erdős-Ko-Rado intersection theorem
zbMATH Keywords: Graham's conjecture
zbMATH Keywords: primes in short intervals
reviewed by: Štefan Porubský
MaRDI profile type: Publication
full work available at URL: s00493-003-0021-4
OpenAlex ID: W2010763265
DOI: 10.1007/S00493-003-0021-4
description: scientific article; zbMATH DE number 2099075