Combinatorics 6: Mutual Sharing Principle

This is the sixth in a series of 7 posts on sharing combinatorics:

Part 1: Combinatorial Principles for DNA Sharing

Part 2: Non-transitivity

Part 3: A Genetic Pigeonhole Principle

Part 4: Transitivity Principles

Part 5: X-Chromosome 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;

or

(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.

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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.