@VinceVatter ... Unless maybe you can rule out the existence of such a set by observing that the set of all triangulations is countable (I think it ought to be but I'm not sure) and therefore its zero-proportion subsets must be finite (again I think this is true but am not sure of it) but if there's even one counterexample to 4CT then there must be an infinite number of them (definitely true; you can always add another region to the map).
But even if that entire chain of reasoning is 100% true, relying on it feels like cheating to me. (This may make more sense if I admit to being not fully on board with the axiom of choice.)
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