5 problems

Let $K_k^n$ be the complete $k$-partite graph with $n$ vertices in each part, let $C_{\geq\ell}$ be the family of cycles of length at least $\ell$, and let $\Omega^{(\ell,k,n)}$ be…

Let $K_4^n$ be the complete $4$-partite graph with $n$ vertices in each part, and let $C_4$ be the cycle of length four. Complete four-partite $C_4$-saturation conjecture. For ever…

Fix $k\geq 2$, let $K_k^n$ be the complete $k$-partite graph with $n$ vertices in each part, and let $F$ be a subgraph of $K_k^n$. Partite saturation limit conjecture. The limit ……

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

Barber–Kühn–Lo–Osthus–Taylor conjecture. For sufficiently large $n$, the minimum-degree bound in the relevant decomposition theorem could be replaced by