Maybe I'm a bit out of touch but there is no such thing as an "Open Relationship".
You are considered friends with someone until either you or them call it exclusive. Then when it becomes exclusive there are limitations put on both members as to dating. And then you're in a relationship.
If you're both dating other people, you're not in a relationship, you're just friends.
Now, if you're talking about an "Open Marriage" that's a whole different can of worms. I would hope that by the 23rd Century people would be more considerate of each others feelings than to do something like that.