This is the sixth in a series of 7 posts on sharing combinatorics:
Part 2: Non-transitivity
Part 3: A Genetic Pigeonhole Principle
Part 4: Transitivity Principles
Part 6: Mutual Sharing Principle
Part 7: Exceptions
Mutual Sharing Principle:
If a number of people all share with one another (that is, if every pair shares), then either:
(a) there is a common strand that all of them share;
(b) there are at most 3 genotypes that all of them fall into; in other words, the people can divided into 3 sets S, T, and U for which the following three statements are true: all the people in S are fully identical to one another, all the people in T are fully identical to one another, all the people in U are fully identical to one another. In case (b), we can also say that at least one-third of the people are fully identical to one another.
Here is a stronger mutual sharing principle for the X chromosome.
Mutual Sharing Principle for the X chromosome:
If a number of people share with one another on a region on the X chromosome and if at least one of them is male, then there is a common strand that all of them share.
If all the people are female, though, then the best we can do is the general version at the beginning of this post.