Can a Pit Card Game be Won if two players refuse to trade with another particular player?
This is actually deceptively simple.  A full set includes every card of that type in the game.  If you possess a card in a specific set, then nobody else can complete that set.  If you possess cards from every set, then no set can be completed without trading with you.

The flip side is that if there is any set that you lack cards from, then that set can necessarily be completed without any trades to you.