Clarkmeister?

[Pov]

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This is for limit Hold'em, when you are out of position, were the last to show strength, and the river puts 4 to a flush on the board when you don't have any of that suit. The move is to bet out and fold to a raise.

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I'll nitpick - the Clarkmeister Theorem doesn't require you have none of the suit nor that you fold to a raise - simply that you bet out when the 4-flush hits and if you're raised you can decide whether to fold, call or re-raise depending upon your actual holding.

I believe the basis for the profitability of this move is that:
a) few hands can raise you
b) some hands that are worse will call you, but wouldn't bet if you checked
c) some hands that are better will fold

Not sure of the OP's question about the turn . . . in any event, Clarkmeister is a highly respected poster on 2+2 and what he writes is often worth reading and thinking about.