Exactly, the program would have never halted if it were possible to colour all triples. Worse even, no one would know why it wouldn't halt... In other words,

Exactly, the program would have never halted if it were possible to colour all triples. Worse even, no one would know why it wouldn't halt... In other words, you have to be lucky to find a counterexample to the given assumption otherwise you're stuck publishing an increased lower bound