There are some combinatorial laws that do apply that might be useful on occasion, including a transitive-like law.
I worked this out quickly; needless to say, if anyone sees any mistakes, please let me know.
In the following, "share" means "share a particular half-identical region with" (all regions are at the same locations).
When a fully identical match is meant in one of these laws, that's spelled out ("fully identical" means fully identical on that same region, of course).
The principles may have to be rephrased for male X chromosomes (and can probably be strengthened for them anyway), since there's only one chromosome involved in that case.
Pigeonhole Principle #1:
If a person shares with 3 other people, then at least 2 of those other people share with each other.
[Looking at the raw data, those two would share a common strand with the original person.]
Pigeonhole Principle #2:
If a person shares with a number of other people, those other people can divided into two sets S and T, where everybody in S shares with one another, and everybody in T shares with one another.
[Looking at the raw data, all the people in S share a common strand with the original person, and all the people in T share the other strand with the original person.]
It's possible for S or T to be empty, or to have just one person in it. In that case, the other set contains all, or all but one, of the other people.
Pigeonhole Principle #3:
If a person shares with N other people, then at least N/2 (rounding up if N is odd) of those other people all share with one another.
Again, all of these N/2 people share a common strand with the original person.
Five-way Transitive Principle:
If person A shares with 5 other people, and if those 5 people all share with person B, then either:
(a) More commonly -- A and B share with each other;
or
(b) More rarely -- at least 2 of the 5 other people share fully identically on the region in question.
N-Way Transitive Principle:
If person A shares with N other people, and if those N people all share with person B, then either:
(a) More commonly -- A and B share with each other;
or
(b) More rarely -- at least N/4 (rounding up) of the N other people share fully identically on the region in question.
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, where:
all the people in S are fully identical to one another,
all the people in T are fully identical to one another,
and all the people in U are fully identical to one another.
("Fully identical" means "fully identical in both strands on the region in question".)
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If 23andMe reports something that seems to violate one of these principles, the apparent anomaly is presumably due to one of the following:
(1) microdeletions;
(2) data errors;
(3) uncertain boundaries on short, close-to-the-threshold matches;
(4) unusual situations where two matches butt up against one another and look like one long match, and this is happening in both strands;
or (5) on the X chromosome, 23andMe's varying threshold sizes. (Male X chromosome results are different anyway, since there's only one of them.)
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Mitchell Spector