7 problems

Asymptotic coloring conjecture. As $s\rightarrow\infty$,

Let $\accentset{\leftrightarrow}{K}_2$ be the complete digraph on two vertices, let $S_2^+$ be the two-out-star, and let $\vec{K}_4^s$ be the unique strong tournament on four verti…

Let $\accentset{\leftrightarrow}{K}_2$ be the complete digraph on two vertices, let $S_2^+$ be the orientation of a two-leaf star with all arcs directed outwards, and let…

An oriented graph is a digraph with no digons, and its dichromatic number $\vec{\chi}(D)$ is the least number of acyclic sets partitioning its vertex set. A digraph is a heroic set…

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 $D$ be a digon-free digraph, meaning that no pair of opposite directed edges joins the same two vertices, and let $\Delta$ be its maximum total degree. Erdős's conjecture. Ther…