[cbloom]

Let me try some numbers.

Say the pot is P on the river. F fraction of the time they have the
flush, 1-F they don't.

If you bet 1/2P , they will always raise with the flush and you will
always fold. If they don't have the flush, they will raise anyway 25% of
the time and you will fold. So, F + 1/4*(1-F) you lose -1/2P , and
3/4*(1-F) you win +P .

EV = (F + 1/4*(1-F))*(-P/2) + 3/4*(1-F)*P

EV = P*[ 5/8 - 9/8F ]

If you check, they will always bet the flush. If they don't have the
flush they'll bet 50% of the time. Assume they bet 3/4P, should you be
calling or folding when they bet?

If you fold to their bet :

EV = 1/2*(1-F)*P
Ev = P * [ 1/2 - 1/2F ]

If you call :

EV = 1/2*(1-F)*P + F * (-3/4P) + 1/2*(1-F)*(7/4P)
EV = P * [ 1/2 - 1/2F - 3/4F + 7/8 - 7/8F ]
EV = P * [ 11/8 - 17/8F ]

if F is 0 (never has flush) the best line is clearly check-call
if F is 1 (always has flush) the bets line is check-fold

if F is 1/2 the EV's are

EV(bet) = P * (5/8 - 9/16) = P/16

EV(check-fold) = P ( 1/2 - 1/4) = P/4

EV(check-call) = P * (11/8 - 17/16 ) = P * 5/16

So the best line is check-call, but check-fold is very close, and betting is much worse

If there any F where betting is best?
Compare to check-call

[ 5/8 - 9/8F ] = [ 11/8 - 17/8F ]
F = 3/4

So betting is better for F >= 3/4