  The iterative absorption method has recently led to major progress in the
area of (hyper-)graph decompositions. Amongst other results, a new proof of the
Existence conjecture for combinatorial designs, and some generalizations, was
obtained. Here, we illustrate the method by investigating triangle
decompositions: we give a simple proof that a triangle-divisible graph of large
minimum degree has a triangle decomposition and prove a similar result for
quasi-random host graphs.
