If property (P) holds for perfect graphs and almost all graphs does it hold for all graphs?
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6$\begingroup$ Perhaps I misunderstood... How about the property "G is perfect or has at least 20 vertices"? $\endgroup$ – Goldstern Sep 21 '12 at 11:31

$\begingroup$ @Goldstern: Obviously this is not the kind of property I had in mind, but you have a good point there. I was thinking of things like Hadwiger's conjecture. $\endgroup$ – Felix Goldberg Sep 21 '12 at 12:11

1$\begingroup$ Take one nonperfect graph of each (large enough) order and define the property as being not equal to that chosen graph. You need to work on your question a lot before it makes sense. $\endgroup$ – Brendan McKay Sep 21 '12 at 12:53

$\begingroup$ Or then again, take any property which have a finite number of exceptions, each of them a non perfect graph. $\endgroup$ – Olivier Sep 21 '12 at 17:30

$\begingroup$ Would some sort of monotonicity (upwards or downwards) constraint on your property make this a meaningful question? $\endgroup$ – D. Ror. Oct 19 '16 at 21:31