Maximal Subsets Free of Arithmetic Progressions in Arbitrary Sets

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.