Geodesic
Maximal Subsets Free of Arithmetic Progressions in Arbitrary Sets
Matematičeskie zametki, Tome 102 (2017) no. 3, pp. 436-444.

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The problem of determining the maximum cardinality of a subset containing no arithmetic progressions of length $k$ in a given set of size $n$ is considered. It is proved that it is sufficient, in a certain sense, to consider the interval $[1,\dots,n]$. The study continues the work of Komlós, Sulyok, and Szemerédi.
Keywords: additive combinatorics, combinatorial number theory. 
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A. S. Semchenkov. Maximal Subsets Free of Arithmetic Progressions in Arbitrary Sets. Matematičeskie zametki, Tome 102 (2017) no. 3, pp. 436-444. http://geodesic.mathdoc.fr/item/MZM_2017_102_3_a8/

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