The key-word is quality, not quantity. You need to stand out of the crowd. Show them what they want to see; devotion, motivation, recommendations etc. Even if you don't have any references, you could always mention that a specific Capt recommended you to apply. Be slick. Use your elbows and get ahead of the queue. Create a convincing story, and point out your ambitions and plans, even if it's not entirely true. You have to sell yourself. As I see it, the phone call is the application, the one in paper is just put on file.
