What do I do when me and a partner break up but my main partner still wants to date them?
That really depends on the specifics of the situation. If all three of you were in a triad, where there’s a single central relationship that bonds you three, then it sounds like that bond has been broken.
But, if you’re in a situation where the dating relationships look like: Adam&Ben, Adam&Carl, and Ben&Carl, then if Adam&Ben break up, then Adam&Carl and Ben&Carl can still continue to date. If Adam & Carl & Ben frequently hang out as a unit of three, that dynamic will necessarily change after a breakup, but the pair relationships don’t need to be all mutually reliant.
Of course, that assumes that the breakup between Adam&Ben was mutual and gentle. If there is drama such that Carl is put off by Adam’s behavior during the breakup, Carl and Adam’s relationship will be affected.
It really depends on your situation. If you and someone broke up, but your other partner still wants to date that person, that might be totally sustainable with some time to cool down and some re-drawn boundaries. If you feel really uncomfortable with it, let your partner know, and talk to them about how to proceed.
On the one hand, it can feel unfair if someone’s relationship is impacted by an interpersonal situation between two other people that they have no part of or control over. On the other hand, some arrangements are just unsustainable, and if your partner decides that they no longer want to date your ex because they are your ex, that’s your partner’s prerogative.