4 problems

Colorability-extension conjecture. For any $r,m,n\in \mathbb{N}$ with $r\ge 3$ and $n\ge (r-1)^2+1$, if all bi-hypergraphs in $Bi\mathcal{H}yp(n,r,m)$ are colorable, then all bi-hy…

Minimum-size conjecture. For any $r\in \mathbb{N}$ with $r\ge 4$,

Conjecture on $\ell(7,3)$.

Existence conjecture. For any $n,r\in \mathbb{N}$ with $r\ge 3$ and $n\ge (r-1)^2+1$, the set $\mathcal{MUC}(n,r)$ is not empty.