Almost disjoint families of 3-term arithmetic progressions
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 \(2\).
scientific article; zbMATH DE number 2136713