Fractional clique decompositions of dense hypergraphs
Michelle Delcourt, Thomas Lesgourgues, Luke Postle · Bulletin of the London Mathematical Society · 2026
Abstract In 2014, Keevash famously proved the existence of ‐Steiner systems as part of settling the Existence Conjecture of Combinatorial Designs (dating from the mid‐1800s). In 2020, Glock, Kühn, and Osthus conjectured a minimum degree generalization: specifically that minimum ‐degree at least suffices to guarantee that every sufficiently large ‐divisible ‐uniform hypergraph on vertices admits a ‐decomposition (where is a constant that is allowed to depend on but not on ). The best‐known progress on this conjecture is from the second proof of the existence conjecture by Glock, Kühn, Lo, and Osthus in 2016 who showed that suffices. The fractional relaxation of the conjecture is crucial to improving the bound; for that, only the slightly better bound of was known due to Barber, Kühn, Lo, Montgomery, and Osthus from 2017. Our main result is to prove that suffices for the fractional relaxation. Combined with the work of Henderson and Postle from 2025, this also shows that such ‐divisible hypergraphs admit ‐decompositions.