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　[Formula　presented].　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　α≥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　[Formula　presented]　if　r　is　even.　©　2024　Elsevier　B.V.