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I think its fine to payoff KK here but once there is a thrid diamond on the river and the number of hands that beat your second set just went up by like a factor of 5(no math done to arrive at this number) I prob just call the raise back to me on the river.

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Maybe I'm missing something, but what are all those additional hands that are beating you? I can only really see AKs, unless he's really crazy about capping the turn with just a flush draw.