New results on a problem of Alon

In 2006, Alon proposed a problem of characterizing all four-tuples $(n,m,s,d)$ such that every digraph on n vertices of minimum out-degree at least s contains a subdigraph on m vertices of minimum out-degree at least d. He in particular asked whether there exists an absolute constant c such that every digraph on 2n vertices of minimum out-degree at least s contains a subdigraph on n vertices of minimum out-degree at least $\frac{s}{2}-c$? Recently, Steiner resolved this case in the negative by showing that for arbitrarily large n, there exists a tournament on 2n vertices of minimum out-degree $s=n-1$, in which the minimum out-degree of every subdigraph on n vertices is at most $\frac{s}{2}-(\frac{1}{2}+o(1\left)\right){\log }_{3}s$.

In this paper, we study the above problem and present two new results. The first result is that for arbitrary large n and any integer $\alpha \ge 2$, there exists a digraph on αn vertices of minimum out-degree $s=n-1$ satisfying that the minimum out-degree of every subdigraph on n vertices is at most $\frac{s}{\alpha }-(\frac{1}{\alpha }+o(1\left)\right){\log }_{\alpha +1}s$. The second result is that for arbitrary large n and any $r\ge 3$, there exists a digraph on 2n vertices of girth r and minimum out-degree s satisfying that the minimum out-degree of every subdigraph on n vertices is at most $\frac{s}{2}-(\frac{1}{2}+o(1\left)\right){\log }_{r}s$ if r is odd, and is at most $\frac{s}{2}-(\frac{1}{2}+o(1\left)\right){\log }_{r+1}s$ if r is even.