---

Claim. For every $r$-uniform hypergraph $F$, every sufficiently large complete $r$-uniform hypergraph $K_n^r$ satisfying the divisibility conditions that $F$ imposes decomposes into edge-disjoint copies of $F$. With $F=K_k^r$ the divisibility conditions are $\binom{k-i}{r-i}\mid\binom{n-i}{r-i}$ for $0\le i<r$ and the decomposition is a Steiner system $S(r,k,n)$, so the answer to Problem 722 is yes for every $k>r$. The authors present the result as a new proof of the existence of block designs, the first being Keevash's existence of designs, and extend it to decompositions of quasirandom and dense hypergraphs. The method, iterative absorption, repeatedly covers almost all edges and reduces the leftover into a small absorbing structure prepared in advance. The memoir is not held in the library; the statement is recorded as the arXiv abstract gives it. The case $F=K_k^r$, which answers the problem, was first posted on 2016-11-21 as arXiv:1611.06827v1; the theorem for arbitrary $F$ was first posted on 2017-06-06 as arXiv:1706.01800, and the arXiv records say the two were merged into arXiv:1611.06827v3, the version that became the memoir.

Acceptance. Refereed: Glock, S., Kühn, D., Lo, A. and Osthus, D., The existence of designs via iterative absorption: hypergraph $F$-designs for arbitrary $F$, Mem. Amer. Math. Soc. 284 (2023), no. 1406, published 2023-03-21; the preprint was first posted 2016-11-21, the date of this page. The site credits the problem to Keevash and does not name this proof, so the page lists no reviewed evidence. No formalization of this proof is recorded; the Lean development linked from Keevash's page names Keevash as its informal author. The proof was not reconstructed in this corpus.