bitter, close, deadly (BrE), fierce, formidable, great, hated, powerful, serious, tough
biggest, chief, largest, leading, main, major, nearest, primary, principal, top (also arch-rival)
long-time, old, traditional
They're old political rivals.

former, one-time
new
The business needed to revive profits and compete with new rivals.

potential
She is now regarded as the greatest potential rival to Hu.

jealous
friendly
Those two have been friendly rivals since they first met.

romantic
They wind up as romantic rivals for the same woman.

foreign, international
The company is well equipped to compete with its international rivals.

business, political
presidential
Conservative, Democratic, etc.

have
She has no rivals for the job.

face
The company faces big rivals in Europe and Asia.

beat, defeat
eliminate
He eliminated his rivals with brutal efficiency.

candidate
clan, faction, gang, group, tribe
company, firm, organization
bid, offer
claim
theory

rival for
They were rivals for her love.

rival in
They were rivals in love.

rival to
Grand it may be, but this cathedral is no rival to the great cathedral of Amiens.

Rival is used with these nouns:
camp, claim, faction, fan, gang, team