4 problems

Let $D$ be a digraph, and let $f4a0$ be the minimum out-degree parameter. For a digraph $F$, define $mader_{\delta^+}(F)$ to be the least integer $m$ such that every digraph $D$ wi…

Let $C$ be an oriented cycle, and let $|V(C)|$ denote its order. Oriented-cycle conjecture. Every digraph $D$ with $\delta^+(D)\geq 2k-1$ contains a subdivision of any oriented cyc…

Let $D$ be a digraph, let $D\langle V_i\rangle$ denote the subdigraph induced by $V_i$, and let $k$ be a positive integer. Alon's partition conjecture. There exists a function $h$…

Let a $\delta^+$-maderian digraph be a digraph contained as a subdivision in every digraph whose minimum out-degree is sufficiently large. An oriented tree is an orientation of an…