The number of wins in Big Fight required for instant promotion depends on the stage you're currently at:

 

Stage 1 - 3 wins

Stage 2 - 9 wins

Stage 3 - 12 wins

Stage 4 - 14 wins (all opponent in your group)

 

If your group doesn't have enough players to get the needed number of wins, Big Fight is still available, but you won't be able to promote instantly.