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I think there's a decent case to check/call with the worst flush draws in these spots, stuff like 8c6c. Basically, when you check/call its quite hard to have a polarized range on turns and rivers because of the nature of what hands are comfortable/profitable to check/call.
Imagine the situation where someone raises preflop, you call in position, it's heads up, he check/calls your bet on the flop, the turn goes check check, and he bets the river. Most regs in that position bet like 2/3 pot and its nearly always value. In reality these players have so little in their range that could for air (basically the weakest ace high hands he c/c'ed), that they should need to bet like 1/6 pot or observant opponents would fold almost 100%. So they essentially need to get to the river with more air, somehow. The problem with this is that the bluffing opportunity simply will not arrive often enough for him to c/c a complete air hand on the flop. Suppose after you bet the flop you'll on average bet the turn 60%, which is not an uncommon frequency. Well that means he'll only have the bluffing opportunity only 40% of the time, in spite of calling the flop 100%.
(assuming flop bet of 1 unit into 1.5 units [2/3 pot], and river bet of 2.5 units into 3.5 units)
0 = (0.4)*x*(2.5) - (0.4)*(1-x)*(3.5) - (0.6)*(1)
x = ~83%
40% of the time he checks turn, 60% of the time he bets turn, of the 40% you'll bet the river and he'll fold x% and call 1-x%. If he folds you win the 1.5 flop pot plus his bet, if he calls you lose the 1 dollar bet plus your 2.5 bluff. And you lose the 1 dollar bet the entire 60% that he bets turn.
He needs to fold 83% of the time. So then in other words you would need to call with only 17% of your range to make him indifferent to the line of c/c flop with complete air.
So we then establish that he has to have some sort of strong equity hand like a flush draw or a gutshot to take this line, which everybody already knows. The problem is that these hands are profitable to c/c AND profitable to bet, and those options are in competition with one another. I've seen people c/c stuff like Kc4c here and IMO it's absolutely gross. The most profitable situation in poker is to have a big pot, a polarized range, and to make a big bet into that pot. Kc4c is a hand that will be able to do this on a huge percentage of board runouts. All strong draws are must bets (or perhaps check/raises) on the flop, even if they have some superficial showdown value like king high or bottom pair, simply because that leads to more situations where you're able to stick your stack in with a range that includes air, and your opponent will feel forced to defend for his stack with medium strength hands.
Even gutshots have this polarity advantage and should probably just be bet on ATx, but it is closer the less nutty and strong the draw is. Which is why I think if you were to c/c a flush draw, it should be a very low one.
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