Hypergraph $F$-designs for arbitrary $F$

TL;DR

This paper proves that for any fixed hypergraph $F$, the necessary divisibility conditions are sufficient for decomposing large complete hypergraphs into copies of $F$, extending classical design theory results.

Contribution

It establishes the existence of $F$-designs for arbitrary $r$-uniform hypergraphs under divisibility conditions, generalizing previous special cases and solving a long-standing problem.

Findings

Sufficient conditions for $F$-designs in large hypergraphs.

Extension of classical design existence results.

Applicability to quasi-random and high minimum degree hypergraphs.

Abstract

We solve the existence problem for $F$-designs for arbitrary $r$-uniform hypergraphs $F$. In particular, this shows 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 $G=K_n^{(r)}$ into edge-disjoint copies of $F$, which answers a question asked e.g. by Keevash. The graph case $r=2$ forms one of the cornerstones of design theory and was proved by Wilson in 1975. The case when $F$ is complete corresponds to the existence of block designs, a problem going back to the 19th century, which was first settled by Keevash. More generally, our results extend to $F$-designs of quasi-random hypergraphs $G$ and of hypergraphs $G$ of suitably large minimum degree. Our approach builds on results and methods we recently introduced in our new proof of the existence conjecture for block designs.