Structures of Diversity

In this talk I will present various criteria of difference in the context of strings, i.e. sequences of symbols taken from a finite or countable set, usually referred to as an alphabet. These problems are motivated by the concept of graph capacity, introduced by Shannon in the context of error-free communication over a noisy channel, where we require the strings to differ in special pairs of symbols depending on the channel. Here we introduce the concept of permutation capacity as a generalization of Shannon’s graph capacity from finite to infinite graphs. We determine the permutation capacity in the case when G is the complement of the infinite path graph. This is somehow surprising considering that most of the problems in this area are unsolved and even the asymptotic solution is unknown. Finally, I will briefly consider problems related to forbiddance relations, which arise when a diversity relation should not hold between the pairs of strings.