I would never introduce my boyfriend and say 'and he's an Oxbridge graduate', where is the need?
I would probably be a bit wary of Oxbridge grads to be honest - I went to a grammar school with 3 other grammar schools in the area, so I know a lot of people who went to Oxbridge, and every single one of them was the kind of quiet, nerdy, bit weird person I wouldn't want to really date. I would equally be put off someone who went to an ex-poly or new university though.