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We address the maximum attainable rate of fingerprinting codes under the marking assumption, studying lower and upper bounds on the value of the rate for various sizes of the attacker coalition. Lower bounds are obtained by considering typical coalitions, which represents a new idea in the area of fingerprinting and enables us to improve the previously known lower bounds for coalitions of size two and three. For upper bounds, the fingerprinting problem is modeled as a communications problem. It is shown that the maximum code rate is bounded above by the capacity of a certain class of channels, which are similar to the multiple-access channel (MAC). Converse coding theorems proved in the paper provide new upper bounds on fingerprinting capacity. It is proved that capacity for fingerprinting against coalitions of size two and three over the binary alphabet satisfies and , respectively. For coalitions of an arbitrary fixed size , we derive an upper bound on fingerprinting capacity in the binary case. Finally, for general alphabets, we establish upper bounds on the fingerprinting capacity involving only single-letter mutual information quantities.