Collinear subsets of lattice point sequences—An analog of Szemerédi's theorem |
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Authors: | Carl Pomerance |
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Institution: | Department of Mathematics, University of Georgia, Athens, Georgia 30602, USA |
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Abstract: | Szemerédi'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. |
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