5 problems

Chen–Yang–Yuan–Zhang conjecture. For $n\geq 3k$ and $s\geq 3$,

Mader's internally-disjoint-path conjecture. Such a system $\mathcal{P}$, set $X$, and partition $\mathcal{Y}$ exist.

Gallai's conjecture. There exists such a system $\mathcal{P}$ and set $X$.

Lovász–Cherkassky conjecture. Under these hypotheses, there exists a system $\mathcal{P}$ of edge-disjoint $T$-paths having the required cut property for every $t\in T$.

Let $k$ be a positive integer, let $G$ be a graph of degeneracy $d$, and let $c_k$ denote a constant depending on $k$. A collection of paths is edge-disjoint if no two paths share…