We will show the following (ZFC) theorem which is recently finished work joint with Joerg Brendle: There is no set of size continuum which is "s_0-shiftable", i.e., which can be translated away from every set in the Marczewski ideal s_0 (where a set of reals is in s_0 if for every perfect set there is a perfect subset disjoint from it).
For regular continuum, the proof is easier, and we will go through it in detail. If there is time left, we will also discuss the singular case.
The theorem is very much in contrast to the respective situation when s_0 is replaced by the meager ideal: there are models (e.g., all models that satisfy CH) with large meager-shiftable (i.e., strong measure zero) sets.