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A necessary condition for the probabilities of a set of events to exhibitBell nonlocality or Kochen-Specker contextuality is that the graph ofexclusivity of the events contains induced odd cycles with five or morevertices, called odd holes, or their complements, called odd antiholes. Fromthis perspective, events whose graph of exclusivity are odd holes or antiholesare the building blocks of contextuality. For any odd hole or antihole, anyassignment of probabilities allowed by quantum mechanics can be achieved inspecific contextuality scenarios. However, here we prove that, for any oddhole, the probabilities that attain the quantum maxima cannot be achieved inBell scenarios. We also prove it for the simplest odd antiholes. This leads usto the conjecture that the quantum maxima for any of the building blocks cannotbe achieved in Bell scenarios. This result sheds light on why the problem ofwhether a probability assignment is quantum is decidable, while whether aprobability assignment within a given Bell scenario is quantum is, in general,undecidable. This also helps to undertand why identifying principles forquantum correlations is simpler when we start by identifying principles forquantum sets of probabilities defined with no reference to specific scenarios.

The quantum maxima for the basic graphs of exclusivity are not reachable in Bell scenarios