The Existence of Designs via Iterative Absorption: Hypergraph F-Designs for Arbitrary F

We solve the existence problem for-designs for arbitrary r-uniform hyper graphs F. This implies that given any r-uniform hypergraph F, the trivially ne-cessary divisibility conditions are sufficient to guarantee a decomposition of any sufficiently large complete r-uniform hypergraph into edge-disjoint copies of F, answers a question asked e.g. by Keevash. The graph case r = 2 was proved Wilson in 1975 and forms one of the cornerstones of design theory. The case F is complete corresponds to the existence of block designs, a problem going to the 19th century, which was recently settled by Keevash. In particular, our argument provides a new proof of the existence of block designs, based on iterative absorption (which employs purely probabilistic and combinatorial methods). Our main result concerns decompositions of hypergraphs whose clique distri-bution fulfills certain regularity constraints. Our argument allows us to employ 'regularity boosting' process which frequently enables us to satisfy these con-straints even if the clique distribution of the original hypergraph does not satisfy This enables us to go significantly beyond the setting of quasirandom hy-pergraphs considered by Keevash. In particular, we obtain a resilience version and decomposition result for hypergraphs of large minimum degree.