In 2014, Keevash proved the existence of $(n,q,r)$-Steiner systems (equivalently $K_q^r$-decompositions of $K_n^r$) 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 $K_q^r$-divisible $r$-graph $G$ on $n$ vertices has minimum $(r-1)$-degree (denoted $δ(G)$ hereafter) at least $\left(1-Θ_r\left(\frac{1}{q^{r-1}}\right)\right) \cdot n$, then $G$ admits a $K_q^r$-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 $δ(G)\ge \left(1-\frac{c}{q^{2r}}\right)\cdot n$ 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 $δ(G)\ge \left(1-\frac{c}{q^{r-1 + o(1)}}\right)\cdot n$ guarantees a $K_q^r$-fractional decomposition.
We prove that for every integer $r\ge 2$, there exists a real $c>0$ such that if a $K_q^r$-divisible $r$-graph $G$ satisfies $δ(G)\ge \max\left\{ δ_{K_q^r}^* + \varepsilon,~~1 -\frac{c}{\binom{q}{r-1}} \right\} \cdot n$, then $G$ admits a $K_q^r$-decomposition for all large enough $n$, where $δ_{K_q^r}^*$ denotes the fractional $K_q^r$-decomposition threshold. Combined with the fractional result above, this proves that $\left(1-\frac{c}{q^{r-1 + o(1)}}\right)\cdot n$ 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.
Proof of Nash-Williams' Intersection Conjecture for countable matroids
证明了如果$ M $和$ N $是公共可数边集$ E $上的有限拟阵，则它们允许公共独立集$I $，使得存在一个二分划$ E=E_{M}\cup E_{N} $，其中$ I\cap E_M $在$ M $中跨越$ E_M $，$ I\cap E_N $在$ N $中跨越$ E_N $.它在可数情形下肯定地回答了Nash-Williams的拟阵交猜想。
On partitioning the edges of an infinite digraph into directed cycles
Nash-Williams证明了无向图$ G $的集合$ E(G) $可以划分为圈当且仅当每个割都有偶数或无限条边。后来C。Rumassen给出了一个更简单的证明，并证明了下面的有向类比定理：边集的有向图可以划分成有向循环当且仅当对于每个子集的顶点的基数的传入和传出的边缘是平等的。本文的目的就是证明这个猜想。
