Snow ball fight? Part of erstwhile hostilities between the Soviet Union and the U.S.! (4,3)

I believe the answer is:

cold war

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But I can't explain it at all! Could you help me understand?

(Other definitions for cold war that I've seen before include "State of tension and hostility" , "Low card (anag.)" , "State of tension after 1945" , "Hostile period after 1945" , "Conflict without engagement" .)

I've seen this clue in The Hindu.

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