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Problems involving arithmetic progressions : ウィキペディア英語版 | Problems involving arithmetic progressions Problems involving arithmetic progressions are of interest in number theory,〔 〕 combinatorics, and computer science, both from theoretical and applied points of view. ==Largest progression-free subsets== Find the cardinality (denoted by ''A''''k''(''m'')) of the largest subset of which contains no progression of ''k'' distinct terms. The elements of the forbidden progressions are not required to be consecutive. For example, ''A''4(10) = 8, because has no arithmetic progressions of length 4, while all 9-element subsets of have one. Paul Erdős set a $1000 prize for a question related to this number, collected by Endre Szemerédi for what has become known as Szemerédi's theorem.
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