The number of wins in Big Fight required for instant promotion depends on the Stage you're currently on:
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 number of players to reach the number of wins, Big Fight is still available, but you won't be able to instantly promote.