Advice about dishonesty

Hey guys, So about six months into my relationship I discovered my boyfriend had been sexting other guys and sending nudes. I confronted him and he was really apologetic. He said it didn’t mean anything but he missed some aspects of being single but didn’t want to break up over it. I said I needed to think about it and eventually told him I didn’t want him doing it and he agreed to stop.A few weeks later I discovered he was at it again and confronted him. He said that he had a problem and found it difficult to stop. Again, we had a long conversation and moved forward. I found out for a third time and got really mad but we talked it through. We had a conversation about the possibility of a more open relationship. This isn’t something I’d ever considered before but I went away and did a lot of research. I came to the conclusion that my fears of an open relationship weren’t totally justified. I agreed, with time, we could have a more open relationship. At first we would have to do stuff together until I had rebuilt my trust and I wanted to know if/when he was sexting with other guys.I thought everything was going really well for the last year or so but recently discovered he’s set up a fake twitter account and has been up to his old tricks. I know it’s easy to say to dump someone when you’re not in love with them but I really would appreciate other perspectives on this.Thanks guys.