What does Sergius bet on in Arms and the Man?
This question refers to events in Act III, when Major Petkoff complains that he can't find his old coat. Catherine, his wife, insists that it's in the blue closet; Petkoff protests that he looked there and didn't find it. Catherine sends her servant, Nicola, to get the coat from the blue closet, and Petkoff responds by proposing a bet: He'll buy her "any piece of jewelry" against a "week's housekeeping money that the coat isn't there." She accepts the wager, and Petkoff, excited, asks if anyone else will bet against him.
Sergius eagerly responds, proposing to bet one of his horses -- "my best charger against an Arab mare for Raina that Nicola finds the coat in the blue closet."
We learn later that Sergius owns twenty horses, so this represents a sizeable bet for him. The choice is also characteristic of the image he wants to project. It's aristocratic, because fine horses are a status symbol of the nobility, and it reminds people of his reputation as a heroic cavalry officer. It suggests family intimacy, because he proposes to submit his winnings to Raina, Petkoff's daughter and Sergius's future wife. And it is chivalrous. Not only is he showing courtesy to the ladies (in pledging his winnings for Raina, and betting in favor of Catherine), the very word "chivalry" is derived from the French for "knight," a horseman warrior.
The one bet in the play is a comic interlude over finding a coat, but it helps to characterize both Bluntschli and Sergius. The bet hinges on the age-old idea that women always know where household items are to be found. Petkoff absolutely insists his jacket is not in the blue closet; his wife insists it is. They make a bet over whether or not the coat is there. Petkoff invites both Bluntschli and Sergius to bet on his side. Bluntschli, the ever pragmatic person, says no: he knows who will win the bet and it is not going to be Petkoff. Sergius, on the other hand, shows his flamboyant need to make the grand gesture without thinking the issue through: he says he will give one of his horses to Raina if the coat is found. Of course it is, and he loses the bet—though we might suspect that the grand gesture is not all that grand, as he plans to marry Raina anyway so, in a sense, he loses nothing.