Fractional Clique Decompositions of Dense Hypergraphs

Abstract

In 2014, Keevash famously proved the existence of $(n,q,r)$-Steiner systems as part of settling the Existence Conjecture of Combinatorial Designs (dating from the mid-1800s). In 2020, Glock, K\"uhn, and Osthus conjectured a minimum degree generalization: specifically that minimum $(r-1)$-degree at least $(1-\frac{C}{q^{r-1}})n$ suffices to guarantee that every sufficiently large $K_q^r$-divisible $r$-uniform hypergraph on $n$ vertices admits a $K_q^r$-decomposition (where $C$ is a constant that is allowed to depend on $r$ but not $q$). The best-known progress on this conjecture is from the second proof of the Existence Conjecture by Glock, K\"uhn, Lo, and Osthus in 2016 who showed that $(1-\frac{C}{q^{2r}})n$ suffices. The fractional relaxation of the conjecture is crucial to improving the bound; for that, only the slightly better bound of $(1-\frac{C}{q^{2r-1}})n$ was known due to Barber, K\"uhn, Lo, Montgomery, and Osthus from 2017. Our main result is to prove that $(1-\frac{C}{q^{r-1+o(1)}})n$ suffices for the fractional relaxation. Combined with the work of R{\"o}dl, Schacht, Siggers, and Tokushige from 2007, this also shows that such hypergraphs admit approximate $K_q^r$-decompositions.