Whenever you have no other tean/rival to cheer for, the rivalry becomes intense.
Auburn - Alabama (no other teams in Alabama)
Utah-BYU (besides the Jazz, there is no other team)
UNC-Duke (two "new" teams in Charlotte so no tradition)
Clemson-USC (nothing else in SC)
For OSU-UM, they have the Bungles, Cavs, Reds, Indians, Tigers, Pistons, Red Wings etc to cheer for, so the rivalry doesn't dominate the sports talk ALL freaking year. Same goes for Miami-FSU-Florida, USC-UCLA, Cincinnati-Xavier, Pitt-WVU.
When THAT game and THOSE teams are the only thing going, the rivalry will absolutely be intense.
Regardless, the biggest rivalries in all of sports is:
New Zealand-Australia in rugby
India-Pakistan in cricket
but NOTHING beats
Brazil-Argentina in soccer!!!!!!!!!!!