  A set of containers for a hypergraph G is a collection of vertex subsets,
such that for every independent (or, indeed, merely sparse) set in G there is
some subset in the collection which contains it. No set in the collection
should be large and the collection itself should be relatively small.
Containers with useful properties have been exhibited by Balogh, Morris and
Samotij and by the authors, along with several applications.
  Our purpose here is to give a simpler algorithm than the one we used
previously, which nevertheless yields containers with all the properties needed
for our previous theorem. Moreover this algorithm produces containers having
the so-called online property, allowing previous colouring applications to be
extended to all, not just simple, hypergraphs.
  For illustrative purposes, we include a complete proof of a slightly weaker
but simpler version of the theorem, which for many (perhaps most) applications
is plenty. We also present applications to the number of solution-free sets of
linear equations, including the number of Sidon sets, announced previously but
not proven.
