Collinear Subsets of Lattice Point Sequences - An Analog of Szemerédi's Theorem

Szemeredi’s theorem states that given any positive number B and natural number k, there is a number n(k, B) such that if n > n(/c, B) and 0 < a, < ... < a, is a sequence of integers with a, < Bn, then some k of the ai form an arithmetic progression. We prove that given any B and k, there is a number m(k, B) such that if m > m(k, B) and ug, u1 ,..., u, is a sequence of plane lattice points with EL, II ui uiwl /I < Bm, then some k of the II* are collinear. Our result, while similar to Szemeredi’s theorem, does not appear to imply it, nor does Szemeredi’s theorem appear to imply our result.