Collinear subsets of lattice point sequences—An analog of Szemerédi's theorem

AbstractSzemerédi'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(k, B) and 0 < a1 < … < an is a sequence of integers with an ⩽ 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 u0, u1, …, um is a sequence of plane lattice points with ∑i=1m…ui − ui−1… ⩽ Bm, then some k of the ui are collinear. Our result, while similar to Szemerédi's theorem, does not appear to imply it, nor does Szemerédi's theorem appear to imply our result