In their monograph on problems in combinatorial number theory, Erdös and Graham ask the following question:
Is it true that if and then there exist or so that and ?
And I can’t find a way to do this where the ‘s are . And, if this indeed is a counterexample, I’m bold enough to conjecture that it’s the easiest. could very well be a candidate too, but I haven’t checked that one thoroughly.
Edit: I now proved that the first one of these is indeed a counterexample, but of course, the proof is a case-by-case-argument (how else could it be?), so I won’t put it here, because of uglyness.