The Alon–Frankl–Huang–Rödl–Ruciński–Sudakov conjecture on fractional matching thresholds

For $k,\ell\in\mathbb N$ with $\ell\leq k-1$, let $c^*_{k,\ell}$ be the smallest number $c$ such that every $k$-graph $H$ on $n$ vertices with

$$\delta_\ell(H)\geq (c+o(1))\binom{n-\ell}{k-\ell}$$

contains a perfect fractional matching, where a perfect fractional matching is a fractional matching of size $n/k$. Alon–Frankl–Huang–Rödl–Ruciński–Sudakov conjecture. For all $\ell,k\in\mathbb N$,

$$c^*_{k,\ell}=1-(1-1/k)^{k-\ell}.$$

This conjecture identifies the asymptotic minimum $\ell$-degree threshold for forcing a perfect fractional matching. The source states that it remains open and is particularly challenging for small values of $\ell$.