Intercollegiate means “between colleges.” You meet it most with sport: an intercollegiate game is one college’s team against another college’s team. In American English, college here often means a university, so think of matches between different 大學. School posters and sports news use the word more than daily chat. Though actually, at old English universities like Oxford, intercollegiate can mean a match between the small colleges inside the same university, not a match against a different university. So the tidy idea is “college versus college,” but which colleges — and whether they belong to one school or two — depends on where you are.