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doi:10.1006/eujc.1997.0176 
Copyright © 1999 Academic Press. All rights reserved.
Regular Article
Intersection Properties of Subsets of Integers
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Tibor Szabó
Received 10 October 1994;
accepted 21 July 1997. ;
Available online 12 April 2002.
Abstract
LetNkbe the maximal integer such that there exist subsetsA1,. . . ,ANk 
{1, 2,
,n} for whichAi ∩ Ajis an arithmetic progression of length at leastkfor every 1 ≤ i < j ≤ Nk. Graham, Simonovits and Sós gave the exact value ofN0. Fork ≥ 2, Simonovits and Sós determined the asymptotic behavior ofNk. In this paper we prove a conjecture of Simonovits and Sós concerning the asymptotic value ofN1. We show that
Moreover, we slightly improve the best-known construction, thus disproving their conjecture on the exact extremal system.
