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. 
@article{MZM_2017_102_3_a8,
     author = {A. S. Semchenkov},
     title = {Maximal {Subsets} {Free} of {Arithmetic} {Progressions} in {Arbitrary} {Sets}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {436--444},
     publisher = {mathdoc},
     volume = {102},
     number = {3},
     year = {2017},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2017_102_3_a8/}
}
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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/