6 problems

Let $T$ be a tournament, let $D$ be a simple digraph on $n$ vertices with no subdigraph isomorphic to $T$, and let $\vec{\alpha}(D)$ denote its largest acyclic set. Strong Erdős–Ha…

Let $D$ be an oriented graph with $n$ vertices and $m$ edges or arcs, and let $\vec{\alpha}(D)$ denote its largest acyclic set. Aharoni–Berger–Kfir conjecture. … For tournaments th…

For each positive integer $n$, let $\vec a(n)$ be the minimum of $\vec\alpha(D)$ over all oriented triangle-free graphs $D$ of order $n$, where $\vec\alpha(D)$ denotes the maximum…

Let $H$ be a tournament. An $H$-free tournament is one that does not contain $H$ as a not necessarily induced subdigraph. For a tournament $T$, write $\vec\alpha(T)$ for its maximu…

Let $H$ be an oriented graph, and let an $H$-free oriented graph be one that does not contain $H$ as a not necessarily induced subdigraph. For an oriented graph $D$, write…

Let $D$ be a simple planar digraph on $n$ vertices. An acyclic set is a vertex set inducing no directed cycle in $D$. Hefetz's conjecture. Every simple $n$-vertex planar digraph ha…