4 problems

Folklore clique decomposition conjecture.

Let $G$ be an $n$-vertex graph, and let the cost vector $\mathbf{c}$ be defined by $c_i=i-1$ for all $i$. Erdős's conjecture. The weighted $2$-clique decomposition number satisfies…

Folklore generalization of Nash-Williams' conjecture. If $n$ is sufficiently large and

Barber–Kühn–Lo–Osthus–Taylor fractional decomposition conjecture. For every $r\geq 3$, there exists $n_0\in\mathbb{N}$ such that, for all $n\geq n_0$, if