  For a countable ordinal epsilon we construct a Sigma^0_2 subset of the Cantor
space for which one may force aleph_epsilon translations with intersections of
size 2i, but such that it has no perfect set of such translations in any ccc
extension. These sets have uncountably many translations with intersections of
size 2i in ZFC, so this answers Problem 3.4 of arxiv:1711.04058 .
