Intersection Properties of Subsets of Integers

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 thatN1 = n22 + O(n5 / 3log3n).Moreover, we slightly improve the best-known construction, thus disproving their conjecture on the exact extremal system.