Maximal collections of intersecting arithmetic progressions
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.
scientific article; zbMATH DE number 2099075