Would you be okay with your boyfriend having alone time (lunch/dinner) with his closest female friend?

I've been with my boyfriend for 8 months, I care about him and we have shared more ups than downs, but this is one thing I cannot seem to be comfortable with. They've known each other 10 years, he claims there's never been anything there. I asked how he knew if they never discussed it and he says... show more I've been with my boyfriend for 8 months, I care about him and we have shared more ups than downs, but this is one thing I cannot seem to be comfortable with.

They've known each other 10 years, he claims there's never been anything there. I asked how he knew if they never discussed it and he says it's just never interested him, being with her. They lived together in her house, but he moved out a few months ago as it made me uncomfortable and he was living there for free to watch her cats.

Things that bother me:
-She's asked him to come along with her fiance and her to pick out a wedding ring. Her bf and her haven't married. Also, her bf lives in another state so she's lonely..
-They went to see the Mayan ruins together and then scuba diving. He never told me this, I found out looking at his online album. When asked who he went with, he said he didn't want to bring up his past.
-She didn't want to meet me last week when she went to pick up her cats he had been watching for a week. She is barely around those cats and he's very attached to them.
-He didn't answer the phone when he was having dinner with her (still daylight) right before she went to pick up the cats. She asked him he says, though they did split the bill.
-She was trying to avoid me, but I got there before she left and met her, no thanks to either of them since he was trying to get me to come after she had left.
-He won't compromise to only meet up with her when I'm there. He says I won't always be available and that it's unreasonable of me. All I'm asking for is a compromise.
-He says he's avoiding her because of me, so does that mean she's trying to hang out with him like old times? I don't know.

So would you stay with someone who refuses to compromise, that is what mainly bothers me. We worked it out and things were good for a few days, but it's still on my mind so I've taken some time away from him to think about it. help?
Update: Sorry it's so long, I got carried away.
Update 2: He told me they were never an item, only friends. They met in college, she was a gf of one of his old friends, but he said they haven't had sex or kissed or even discussed that road. I don't know how you would know unless you bring it up, right?
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