If you're looking for the relationship moslty as love love, have a look here, please! :D

    I'm asking myself why I'm writing this section! XD It's because watching Umineko (both anime or game, obviously), it seems clear.

    Beatrice and Battler are rivals from the first time they see each others, in fact he's the one chosen by her to become his rival and grant her wish. Their relationship starts because of this: he's the only one who does not accept her like a witch since he's too rational and doesn't believe in magic. Because of this fact, Beatrice understands that he's the only one who can try to solve the ridle of the epigraph or, anyway, to beat her.

   The most of the times, it is Beatrice the one who mocks Battler and tries to challenge him, but he doesn't give up and always accepts her challenges. Battler always tries to find a solution that doesn't imply magic, even if it can be impossible even for a human being. To avoid this, Beatrice, starting from the second game, will be integral part of the game, but Battler will still continue to think and to prove with nonsense suppositions that it is not about magic, all the deaths are due to other persons, even if he doesn't know who are these other ones and can't see them. XD

   Battler will start to hate Beatrice when he will be in Purgatorio with her and won't be just a pawn, but her opponent. He doesn't want to see his relatives dying and this is why he will have a fight many times with her. From her side, Beatrice always enjoys his point of view and lols as a crazy girl when he says a strange and mystical supposition. She will start to think about Battler as a toy, then he will become her rival and at the end he will be something more. Even if Battler knows very well that the game will continue as long as the gold won't be found, he will continue through his path, without pull back, even if there are problems, he will go on and overtake them.

   To help Battler to understand that there is always magic behind all, Beatrice will start to say sentences written in red (red truth) that are only the truth, they can't be lies, or when she tells them, she will die. The same goes for Battler with his blue truth, with which he makes questions the most of the times. The interesting thing is that both of them can avoid of replying to the questions and Beatrice does it when she can't reply with something that could help him too much.

   We see the importance of this rivalry when Beatrice finds out that Battler gets mad and doesn't want to have her as opponent. She will start to crey and will leave to Ronove her place, but just for a bit, because she is too much fond of their match.