Maximal subsets free of arithmetic progressions in arbitrary sets

We consider the problem of determining the maximum cardinality of a subset containing no arithmetic progressions of length $k$ in a given set of size $n$. It is proved that it is sufficient, in a certain sense, to consider the interval $[1,\dots ,n]$. The study continues the work of Koml'os, Sulyok, and Szemer'edi.