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If C is a class of sets and A is not in C, then an infinite set H is a proper hard core for A with respect to C, if H ⊆ A and for every C &egr; C such that C ⊆ A, C &Cap; H is finite. It is shown that if C is a countable class of sets of strings that is closed under finite union and finite variation, then every infinite set not in C has a proper hard core with respect to C. In addition, the density of such generalized complexity cores is studied.

The existence and density of generalized complexity cores