My denials have gotten me nowhere. They upset him as he thinks I am lying to him
He may be making accusations because he becomes momentarily gripped with a vague but intense source of distrust. He fails to locate where this distrust comes from in himself, so he tries to find a source outside himself.
Trust is particularly hard for someone suffering from BPD.
There is a desire to merge with you (engulfment), equally strong, there is a drive to be separate (abandonment). Both cause interpersonal distress, but he can't find the middle ground between the two.
Because the distrust/fear is based in a feeling, and not fact, you have to respond to the feeling, without confirming the (inaccurate) fact.
It must be very distressing for him to feel that you are having an affair -- this is a feeling you can validate.
If he becomes too dysregulated when validating that feeling, you bookend it with the truth. Many of us race right to the truth, finding it difficult to empathize with such profound distrust.
If you find that validating the feelings creates a bigger dysregulation, you will need to have a boundary about the discussion, and stick to it. Your boundary might be that you walk away if he begins to call you names or blows up into a rage. Best to come up with a boundary together, collaboratively, when he's emotionally regulated, if you have that capacity in your relationship.
Sometimes, a person with BPD will run up and down an emotional register, kind of flitting in and out between different age/states. You might find that you have to be a little more authoritative when your loved one thinks you're having an affair.
A lot of the change will come from your own desire to learn and adapt and grow.

LnL