Still not a sufficient explanation but I pretty much like the headline:
Mathematicians are finding inevitable structures in sufficiently large sets of integers.
Mathematicians are finding inevitable structures in sufficiently large sets of integers.
all I can tell you (because it's all I know) is that "the structure" is preserved by transformations.
and what the subject focuses on and studies is the transformations, more than 'the structures'.
so I suppose they don't precisely and explicitly know what 'the structure' is; but the incredible thing is how this does NOT matter.
as far as I've figured out so far, the point is that it gets preserved across transformations, and that they can inter-relate it across different mathematical objects.