The existence of designs via iterative absorption: hypergraph $F$-designs for arbitrary $F$

Understand

We solve the existence problem for $F$-designs for arbitrary $r$-uniform hypergraphs~$F$.

• This implies that given any $r$-uniform hypergraph~$F$, the trivially necessary divisibility conditions are sufficient to guarantee a decomposition of any sufficiently large complete $r$-uniform hypergraph into edge-disjoint copies of~$F$, which answers a question asked e.g.~by Keevash.
• The graph case $r=2$ was proved by Wilson in 1975 and forms one of the cornerstones of design theory.
• The case when~$F$ is complete corresponds to the existence of block designs, a problem going back to the 19th century, which was recently settled by Keevash.