Collocations for rivalry - noun

big, bitter, fierce, great, heated, intense
the bitter rivalry that existed between them

ancient, long-standing, old, traditional
friendly
petty
international, local, regional
internal, inter-service (esp. AmE)
sibling
ethnic, personal, political, professional, religious
Bitter ethnic rivalries within the region have grown in recent years.

football

grow

rivalry among, rivalry between
the increasing rivalry among competitors in the industry
rivalry between the army and the police

rivalry for
rivalry for the party leadership

rivalry in
the fiercest rivalry in professional sports

rivalry over
rivalry over who is to head the delegation

rivalry with
rivalry with foreign companies