8 problems

Let $a_n$ be the number of acyclic directed graphs with $n$ labeled vertices, and let $b_n$ be the number of real $n\times n$ matrices with entries in $\{0,1\}$ whose eigenvalues a…

Morell and Skutella's conjecture. Any fractional flow $(x_e)_{e\in E}$ can be expressed as a convex combination of unsplittable flows $(y_e)_{e\in E}$ satisfying both bounds. The s…

Let $\overleftrightarrow{r_3}(H)$ be the three-color directed Ramsey number, and let an acyclic digraph have maximum degree at most $\Delta$. Three-color directed-Ramsey conjecture…

For $k,\Delta\geq 1$ and $n>\Delta$, let $H_{k,n,\Delta}$ be an acyclic digraph with $n$ vertices and maximum degree $\Delta$ maximizing $\ovar{r_k}(H)$. Quantitative growth conjec…

For $k,\Delta\geq 1$ and $n>\Delta$, let $H_{k,n,\Delta}$ be an acyclic digraph with $n$ vertices and maximum degree $\Delta$ maximizing $\ovar{r_k}(H)$. Super-polynomial growth co…

Let $D$ be a countable acyclic digraph, and assign to each vertex $v$ a list $L(v)$ of available colors. The digraph is majority $k$-choosable if, for every such assignment with…

Let $D$ be a countable acyclic digraph. A vertex coloring of $D$ is a majority coloring if, at every vertex, at most half of its outgoing edges are bad, meaning that their endpoint…

Let $G$ be a bounded labeled acyclic digraph with a balanced linear edge labeling. Non-negativity conjecture. The $\mathbf{c}\mathbf{d}$-index of $G$ is non-negative. This conjectu…