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In this thesis we consider, within a unified framework, several problems inspired by generalizations of the concept of graph capacity, introduced by Shannon in 1956 in the context of error–free communication over a noisy channel. Along the lines of the various generalizations of capacity we introduce and study diversity relations between combinatorial structures like strings over a discrete (finite or infinite) alphabet or strings of vertices of graphs (or hypergraphs). We consider a binary relation over a set of combinatorial objects a diversity relation if it is irreflexive (meaning that no object is in this relation with itself) and local, i.e. for two objects having projections (induced substructures) in this relation, the objects themselves are in this relation. Moreover, we consider and analyze forbiddance relations, which arise when a diversity relation should not hold between the pairs of objects. In case of both types of these relations we study the largest cardinality of sets of strings whose any pair (k–tuple) of elements is in the same relation