2025/12/03 by Henderson, Cicely, Postle, Luke
#05B05 #05B07 #05B40 #05C35 #05C65 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2512.04071
In 2014, Keevash proved the existence of (n,q,r)-Steiner systems (equivalently Kqr-decompositions of Knr) for all large enough n satisfying the necessary divisibility conditions. In 2021, Glock, Kühn, and Osthus proposed a generalization of this result. Namely they conjectured a hypergraph version of Nash-Williams' Conjecture positing that if a Kqr-divisible r-graph G on n vertices has minimum (r-1)-degree (denoted δ(G) hereafter) at least (1-Θr(\frac1qr-1)) ⋅ n, then G admits a Kqr-decomposition. The best known progress on this conjecture dates to the second proof of the Existence Conjecture by Glock, Kühn, Lo, and Osthus wherein they showed that δ(G)≥ (1-\fraccq2r)⋅ n suffices for large enough n, where c is a constant depending on r but not q. As for the fractional relaxation, the best known bound is due to Delcourt, Lesgourgues, and the second author, who proved that δ(G)≥ (1-\fraccqr-1 + o(1))⋅ n guarantees a Kqr-fractional decomposition. We prove that for every integer r≥ 2, there exists a real c>0 such that if a Kqr-divisible r-graph G satisfies δ(G)≥ max\ δKqr^* + ε,~~1 -\fracc\binomqr-1 \ ⋅ n, then G admits a Kqr-decomposition for all large enough n, where δKqr^* denotes the fractional Kqr-decomposition threshold. Combined with the fractional result above, this proves that (1-\fraccqr-1 + o(1))⋅ n suffices for the Hypergraph Nash-Williams' Conjecture, approximately confirming the correct order of q. Our proof uses the newly developed method of refined absorption; we also develop a non-uniform Turán theory to prove the existence of many embeddings of absorbers which may be of independent interest.