On multiplicative representations of integers

Let 1 ≦ a 1 a k ≦ x ; b 1 b 1 ≦ x . Assume that the number of solutions of a 1 b 1 = m is less than c . The authors prove that then . They also give a simple proof of Szemeredi's theorem: If the products a i b j are all distinct then . They conjecture that (2) holds for c 2 = 1 + e if x > x 0 (e). Several other unsolved problems are stated.