7 problems

Fractional Nash-Williams' conjecture. If

For integers $n\geq r\geq 2$, a fractional $(r-1,r,n)$-Steiner system in an $r$-uniform hypergraph $H$ is a function $\varphi:E(H)\to\mathbb{R}_{\geq0}$ such that…

Let $\theta_{\operatorname{STS}}$ denote the minimum codegree threshold for Steiner triple systems, and let $\theta_{\operatorname{STS}}^*$ denote the corresponding fractional thre…

Threshold equality conjecture. For all $r>k\ge 2$,

Let $K_k^n$ be the complete $k$-uniform hypergraph on $n$ vertices, and expose its edges one by one in a uniformly random order to obtain the random hypergraph process. A facet is…

Fractional-decomposition threshold conjecture. The threshold for the appearance of fractional $K_k$-decompositions in $\mathcal{H}_k(n;p)$ is

Let $G$ be a graph with $n$ vertices and minimum degree at least $\frac{3}{4}n$. Garaschuk's conjecture. If $n$ is large enough, then $G$ is fractionally $K_3$-decomposable. Fracti…