  A hypothetical exclusion principle for quantum particles is introduced that
generalizes the exclusion and inclusion principles for fermions and bosons,
respectively: the correlated exclusion principle. The sum-free condition for
Schur numbers can be read off as a form of exclusion principle for quantum
particles. Consequences of this interpretation are analysed within the
framework of quantum many-body systems. A particular instance of the correlated
exclusion principle can be solved explicitly yielding a sequence of quantum
numbers that exhibits a fractal structure and is a relative of the
Thue-Thurston sequence. The corresponding algebra of creation and annihilation
operators can be identified in terms of commutation and anticommutation
relations of a restricted version of the hard-core boson algebra.
